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The Single Server Queue with Catastrophes and Geometric Reneging
Abstract
We consider the single server queue with catastrophes that occur according to a Poisson process. The catastrophes force all present customers to abandon the system and render the server inoperative. Then a repair time is set on till the server becomes ready to serve again the customers. In the meanwhile the customers are accumulated according to their arrival process. Recently, several authors have investigated the reneging behavior in such systems and other related models when the customers become impatient because of the failure of the server. Two kinds of reneging have been considered: independent and binomial. In the case of independent reneging, each customer has its own patience time and abandons the system as soon as it expires. In the case of binomial reneging, the abandonment opportunities occur according to a certain point process and then all present customers decide simultaneously but independently whether they will abandon the system or not. In the present paper, we complement these studies by considering the case of geometric reneging. This case arises when the abandonment opportunities occur according to a certain point process and the customers decide sequentially whether they will leave the system or not. We derive explicit expressions and computational schemes for various performance descriptors, concerning the number of customers in system, the sojourn time of a customer, the duration and the maximum number of customers in a busy period.
... The study of the number of customers in system reduced according to a geometric distribution is a recent endeavor. For more detailed and excellent studies on this policy, the readers may refer to Artalejo et al. [1], Economou and Gomez-Corral [8], Dimou et al. [6,7]. For instance, in [8], the authors dealt with a population of individuals that grows stochastically according to a batch Markovian arrival process and is subject to renewal generated geometric catastrophes. ...
... In [7], the authors considered a single server vacation queueing model, where the customers become impatient during the absence of the server, and abandon the system according to a geometric distribution. Then, in [6], the authors studied the single server queue with catastrophes and geometric reneging, where the customers become impatient and leave the system according to a geometric distribution while the server is in repair. In fact, the mechanism for the geometric abandonment is well-motivated by applications in various fields, especially in manufacturing systems and perishable inventory systems. ...
... Then, all present customers start sequentially to leave the system and the reduction of the number of customers ceases at the first individual who determines to stay in the system, or when all present customers abandon it. For more details, we may refer interested readers to [6,7]. This abandonment policy can be interpreted as a strategy that at a server breakdown epoch the number of customers in the system is decreased according to a geometric distribution. ...
This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e., an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phase i with probability qi, i = 1, 2, ..., d. Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.
... Hereafter we are able to state the key result of the paper. We prove that, under suitable assumption, the probability distribution q 0,n (t) = P {M (t) = n|M (0) = 0} (16) can be expressed in terms of the probability distribution of [N (t)|N (0) = 0] and the distribution of Z. ...
... This result confirms that the distribution (8) of Z can be viewed as the steady-state distribution of the birth-death process N (t) subject to catastrophes, here denoted as N c (t). As a consequence, we are now able to provide an alternative form for (16) in terms of the conditional probabilities of N c (t). ...
... Under the assumptions of Proposition 1, for t > 0 the probabilities (16) can be also expressed as: ...
For a birth-death process N(t) with a reflecting state at 0 we propose a method able to construct a new birth-death process M(t) defined on the same state-space. The birth and death rates of M(t) depend on the rates of N(t) and on the probability law of the process N(t) evaluated at an exponentially distributed random time. Under a suitable assumption we obtain the conditional probabilities, the mean of the process, and the Laplace transforms of the downward first-passage-time densities of M(t). We also discuss the connection between the proposed method and the notion of -similarity, as well as a relation between the distribution of M(t) and the steady-state probabilities of N(t) subject to catastrophes governed by a Poisson process. We investigate new processes constructed from (i) a birth-death process with constant rates, and (ii) a linear immigration-death process. Various numerical computations are performed to illustrate the obtained results.
... Recently, [12] presented the transient analysis of impatient customers in a Markovian single server queue with disasters queue in random environment. Other contributions on impatience customers' in different queueing models can be found in [13,14,15,16,17,18,19,20,21]. ...
... Next, differentiating Eq. (14) and taking z = 1, we find: ...
Machining systems are essential for many industrial applications, such as manufacturing, processing, and assembly. However, these systems are often exposed to various sources of uncertainty and disruption, such as disasters and customer impatience. These factors can adversely affect the performance, reliability, and profitability of the machining systems. Consequently, modeling and analyzing machining systems under these conditions becomes crucial. In this paper, we deal with a Markovian multi-server queueing system with batch arrival, Bernoulli feedback, and customers? impatience (balking and reneging). The system undergoes disastrous interruptions that force all customers- whether waiting or currently in service-to exit, leading to server failures. Moreover, the system dynamically alternates between main servers and substitute servers based on the occurrence of disasters. These substitute servers operate at reduced rates compared to the main servers. Our contributions include deriving the stability condition for the system and obtaining the probability generating function of steady-state probabilities, enabling us to derive essential performance measures. Additionally, we develop a cost model and conduct an economic analysis for the system.
... Yechiali [16] analysed queue with disaster and impatience. Sudesh [13], Dimou and Economou [3] were some of the remarkable papers in queue with disasters and impatience. ...
... Also the denote the time dependent probability for the system to be in state with customers at time . Assume that initially the system is empty and the server is being idle ie., 3,0 (0) 1. P = By standard methods, the system of Kolmogorov differential difference equations governing the process are given by ...
An / /1 queueing model with disasters and repairs under Bernoulli working vacation schedule is considered. In this model, after every completion of service the server may take vacation with probability or the server may render service to the next customer with probability. By considering the disaster to occur, only when the server is in busy state, the explicit analytical expressions for time dependent probabilities are derived using Laplace transform and generating function technique.
... They called this type of abandonment as synchronized or binomial abandonment. A third type of reneging, known as sequential or geometric abandonment, was presented by Dimou et al. [6], Dimou and Economou [5]. Both synchronized and sequential abandonments are incited by remote systems, where users have to look for the arrival of the secondary transport facility (with different capacity depending on the type of abandonment) to abandon the system. ...
... Let N(t) be the number of customers in the system at time t, and ζ(t) = 0 if the server is on inactive mode (vacation), 1 if the server is in active mode (busy). [5] Strategies with vacations and synchronized abandonment 5 ...
We study impatient customers’ joining strategies in a single-server Markovian queue with synchronized abandonment and multiple vacations. Customers receive the system information upon arrival, and decide whether to join or balk, based on a linear reward-cost structure under the acquired information. Waiting customers are served in a first-come-first-serve discipline, and no service is rendered during vacation. Server’s vacation becomes the cause of impatience for the waiting customers, which leads to synchronous abandonment at the end of vacation. That is, customers consider simultaneously but independent of others, whether to renege the system or to remain. We are interested to study the effect of both information and reneging choice on the balking strategies of impatient customers. We examine the customers’ equilibrium and socially optimal balking strategies under four cases of information: fully/almost observable and fully/almost unobservable cases, assuming the linear reward-cost structure. We compare the social benefits under all the information policies.
... Ammar et al. [4] investigated the transient solution for a single server queuing system with impatient customers using matrix approaching. An M/M/1 queueing system with catastrophes and geometric reneging was analyzed by Dimou and Economou [15]. They obtained the steady-state expressions for system size probabilities and some performance measures. ...
... Clearly, these transient and steady-state probabilities agree with Eqs. (10), (15) and (20) of Krishna Kumar and Pavai Madheswari [26], respectively. ...
An M / M / 1 queue with reneging, catastrophes, server failures and repairs is considered. The arrivals follow a Poisson process and the servers serve according to an exponential distribution. On arrival a customer decides to join the queue and after joining the queue if a customer has to wait for the service longer than his expectation, he may renege. Explicit expression for the time-dependent probabilities of the system size is obtained in terms of the modified Bessel function of first kind by making use of Laplace transform and probability generating function techniques. The system queue length and failure distribution for steady state are derived. Additionally, time-dependent mean and variance are obtained. A numerical example is presented to study the behavior of the system.
... We include a repair mechanism in the queueing system under investigation, since it is essential to model instances when the (random) repair times are not negligible. We remark that the interest in this feature is increasing in the recent literature on queueing theory (see, for instance, Dimou and Economou [25]). ...
... The integrand on the right-hand side of Equation (25) refers to the sample-paths of N(t) that start from zero at time t 0 , then reach the state zero at time τ ∈ (t 0 , t) and, finally, go from zero at time τ to n at time t for the first time, without the occurrence of catastrophes in (τ, t). ...
... We include a repair mechanism in the queueing system under investigation, since it is essential to model instances when the (random) repair times are not negligible. We remark that the interest in this feature is increasing in the recent literature on queueing theory (see, for instance, Dimou and Economou [25]). ...
... The integrand on the right-hand side of Equation (25) refers to the sample-paths of N(t) that start from zero at time t 0 , then reach the state zero at time τ ∈ (t 0 , t) and, finally, go from zero at time τ to n at time t for the first time, without the occurrence of catastrophes in (τ, t). ...
We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by a time-varying intensity function. We analyze the transient and the asymptotic behavior of the queueing system. Moreover, we derive a heavy-traffic approximation that allows approximating the state of the systems by a time-non-homogeneous Wiener process subject to jumps to a spurious state (due to catastrophes) and random returns to the zero state (due to repairs). Special attention is devoted to the case of periodic catastrophe and repair intensity functions. The first-passage-time problem through constant levels is also treated both for the queueing model and the approximating diffusion process. Finally, the goodness of the diffusive approximating procedure is discussed.
... We include a repair mechanism in the queueing system under investigation, since it is essential to model instances when the (random) repair times are not negligible. We remark that the interest in this feature is increasing in the recent literature on queueing theory (see, for instance, Dimou and Economou [25]). ...
... The integrand on the right-hand side of Equation (25) refers to the sample-paths of N(t) that start from zero at time t 0 , then reach the state zero at time τ ∈ (t 0 , t) and, finally, go from zero at time τ to n at time t for the first time, without the occurrence of catastrophes in (τ, t). ...
... Kapodistria, Phung-Duc, and Resing [12] investigated a birth/immigration-death processes under binomial catastrophes and gave an elaborate analysis on both the transient and equilibrium distribution of the population size. For some related papers on impatient customers and diasters, the interested readers are referred to Yechiali [23], Chakravarthy [4], Sudhesh [19] and Dimou and Economou [5], and the references therein. Lately, the analysis of such queueing systems with disasters/catastrophes focuses on an economic viewpoint, see, for example, Economou and Manou [7], Boudali and Economou [2,3]. ...
... In this section, similarly to the method used in Dimou and Economou [5] and Dimou, Economou, and Fakinos [6], we study the busy period of the system, which is defined as the time from the epoch at which an arriving customer finds the system empty to the next epoch that the system is empty again (due either the server having accumulated N negative feedbacks or due to the server having served all present customers). ...
This paper is devoted to the study of a clearing queueing system with a special discipline. As soon as the server receives N negative feedbacks from customers, all present customers are forced to leave the system and the server undergoes a maintenance procedure. After an exponential maintenance time, the system resumes its service immediately. Using the matrix analytic method, we derive the steady-state distributions, which are then used for the computation of other performance measures. Furthermore, using first step analysis, we obtain the Laplace–Stieltjes transform of the sojourn time of an arbitrary customer. We also study the busy period of the system and derive the generating function of the total number of lost customers in a busy period. Finally, we investigate a long-run rate of cost and explore the optimal N value that minimizes the total cost per unit time. We also present some numerical examples to illustrate the impact of several model parameters to the performance measures.
... Switching Failure Kuo and Ke [36], Shekhar et al. [43], Hsu et al. [75], Jain and Preeti [78], Jain and Rani [79], Jain et al. [81], Jain et al. [100], Jain et al. [112], Ke et al. [122], Jain et al. [139]. Common Cause Failure Mechri et al. [61], Jain and Gupta [76], Dimou and Economou [92], Jain [99], Jain et al. [100], Jain and Gupta [109], Jain et al. [139], Maheshwari et al. [146]. Degraded Failure ...
... RenegingMadheswari et al.[38], Yang et al.[68], Ammar et al.[91], Jain et al.[112], Maheshwari et al.[145]. Geometric Reneging Shekhar et al.[41], Shekhar et al.[43], Dimou and Economou[92].Balking Ammar et al.[91], Jain et al.[112]. ...
The aim of the present article is to give a historical survey of some important research works related to qnenes in machining system since 2010. Qnenes of failed machines in machine repairing problem occur dne to the failure of machines at random in the manufacturing industries, where different jobs are performed on machining stations. Machines are subject to failure what may result in significant loss of production, revenue, or goodwill. In addition to the references on qnenes in machining system, which is also called 'Machine Repair Problem' (MRP) or 'Machine Interference Problem7 (M1P). a meticulous list of books and snrvey papers is also prepared so as to provide a detailed catalog for understanding the research in qneneing domain. We have classified the relevant literature according to a year of publishing, methodological, and modeling aspects. The anthor(s) hope that this snrvey paper conld be of help to learners contemplating research on qneneing domain.
... Switching Failure Kuo and Ke [36], Shekhar et al. [43], Hsu et al. [75], Jain and Preeti [78], Jain and Rani [79], Jain et al. [81], Jain et al. [100], Jain et al. [112], Ke et al. [122], Jain et al. [139]. Common Cause Failure Mechri et al. [61], Jain and Gupta [76], Dimou and Economou [92], Jain [99], Jain et al. [100], Jain and Gupta [109], Jain et al. [139], Maheshwari et al. [146]. Degraded Failure ...
... RenegingMadheswari et al.[38], Yang et al.[68], Ammar et al.[91], Jain et al.[112], Maheshwari et al.[145]. Geometric Reneging Shekhar et al.[41], Shekhar et al.[43], Dimou and Economou[92]. BalkingAmmar et al.[91], Jain et al.[112]. ...
The aim of the present article is to give a historical survey of some important research works related to queues in machining system since 2010, queues of failed machines in machine repairing problem that occur due to the failure of machines at random in the manufacturing industries, where different jobs are performed on machining stations. Machines are subject to failure which may result in significant loss of production, revenue, or goodwill. In addition to the references on queues in machining system which is also called `Machine Repair Problem' (MRP) or `Machine Interference Problem' (MIP), a meticulous list of books and survey papers is also prepared so as to provide a detailed enough catalog for an absolute understanding of research in queueing domain. We have classified the relevant literature according to a year of publishing, methodological, and modeling aspects. The author(s) hope that this survey paper could be of help to learners contemplating research on queueing domain.
... Informally, following [1], a random process with catastrophes can be represented as the difference between two components: a regular component and a catastrophic one. These processes serve as a tool for understanding and predicting various phenomena across diverse fields such as ecology and biology ( [2], [3], [4], [5]), economics (see [6], [7], [8]), queueing systems ( [9], [10], [11], [12], [13]), and more. Much literature exists on processes involving catastrophes, and we do not claim to provide a comprehensive list of relevant references. ...
In this paper, we expand and generalize the findings presented in our previous work on the law of large numbers and the large deviation principle for Poisson processes with uniform catastrophes. We study three distinct scalings: sublinear (moderate deviations), linear (large deviations), and superlinear (superlarge deviations). Across these scales, we establish different yet coherent rate functions.
... These studies have provided a foundation for exploring more complex abandonment patterns, such as geometric and synchronized abandonment. Dimou and Economou (2013) analyzed the M/M/1 queue with geometric reneging and catastrophes, providing valuable insights into the impact of geometric abandonment and unexpected system failures. The concept of synchronized abandonment, involving various variations in simple Markovian queues, has been thoroughly discussed in the works of Economou (2004), Adan et al. (2009), Economou and Kapodistria (2010), Kapodistria (2011), Panda et al. (2016) and Panda and Goswami (2020). ...
We investigate the dynamics of customer geometric abandonment within a queueing framework characterized by renewal input batch arrivals and multiple vacations. Customers’ impatience becomes evident when confronted with server vacations, triggering instances of abandonment. This phenomenon reduces the number of customers within the system during abandonment epochs following a geometric distribution. The probability of customers leaving the queue escalates with prolonged waiting times. We derive concise and closed-form expressions for system-length distributions at pre-arrival and arbitrary epochs by harnessing the power of supplementary variable and difference operator methods. Furthermore, we elucidate specific instances of our model, shedding light on its versatility. To substantiate our theoretical framework, we provide a series of illustrative numerical experiments presented through meticulously crafted tables and graphs, thereby showcasing the robustness and applicability of our methodology.
... Total catastrophes, which happen occasionally and make the population disappear, have an impact on this process. The effects of different types of catastrophes on population processes had been examined by Granita [1], Getz [2], E.G. Kyriakidis et al. [4], Michael et al. [8], and Sindayigaya [3].Some researchers discuss the recurring and progressive nature of the birth and death processes in random environments [8][9][10].The main objective of this paper is to examine the BDI process using a catastrophe parameter and based on the DDE. Using a PGF, the general solution is obtained, which ultimately results in the calculating the population's mean and variance and, in the end, yields numerical examples. ...
This paper, delineates how catastrophe affect the Birth-Death and Immigration Process (BDI). To find the general solution for the population size distribution, the Probability Generating Function (PGF) is used. When the rates of birth, death, immigration, and catastrophe are remain's constant, the accurate solution of mean and variance is also obtained. In addition, the mean and variance are time dependent. The system's behavior, a numerical example is presented for investigated.
... The analysis of reset/catastrophe processes is becoming more and more appealing, since they play a relevant role in various applied contexts, such as queueing, population dynamics, stochastic modeling and physics. In this framework, birth-death processes and queueing systems subject to catastrophes have been studied in Di Crescenzo et al. [16], Boudali and Economou [6], Dimou and Economou, [21] and Giorno et al. [27], among others. Moreover, in Economou and Gómez-Corral, [23], the authors study the influence of renewals generated by geometric catastrophes on a population of individuals that grows stochastically according to a batch Markovian arrival process (BMAP). ...
We investigate the effects of the resetting mechanism to the origin for a random motion on the real line characterized by two alternating velocities v 1 and v 2 . We assume that the sequences of random times concerning the motions along each velocity follow two independent geometric counting processes of intensity λ , and that the resetting times are Poissonian with rate ξ > 0 . Under these assumptions we obtain the probability laws of the modified telegraph process describing the position and the velocity of the running particle. Our approach is based on the Markov property of the resetting times and on the knowledge of the distribution of the intertimes between consecutive velocity changes. We obtain also the asymptotic distribution of the particle position when (i) λ tends to infinity, and (ii) the time goes to infinity. In the latter case the asymptotic distribution arises properly as an effect of the resetting mechanism. A quite different behavior is observed in the two cases when v 2 < 0 < v 1 and 0 < v 2 < v 1 . Furthermore, we focus on the determination of the moment-generating function and on the main moments of the process describing the particle position under reset. Finally, we analyse the mean-square distance between the process subject to resets and the same process in absence of resets. Quite surprisingly, the lowest mean-square distance can be found for ξ = 0 , for a positive ξ , or for ξ → + ∞ depending on the choice of the other parameters.
... Most of the analysis of queuing systems, here, have focused on one type of disaster, only on busy period with repair state, for instance: the analysis of a single-server queue with disasters and repairs under Bernoulli vacation schedule [10]; the study of queues with impatient customers and disasters when the system is down [11]; the analysis of a disaster queue problem with Markovian and impatient arrivals [3]; and the Markovian single queue with disasters and geometric reneging [4]. ...
This paper studies the stationary analysis of a Markovian queuing system with Bernoulli feedback, interruption vacation, linear impatient customers, strong and weak disaster with the server's repair during the server's operational vacation period. Each customer has its own impatience time and abandons the system as soon as that time ends. When the queue is not empty, the server's operational vacation can be interrupted if the service is completed and the server starts a busy period with a probability q or continues the operational vacation with a probability q. A strong disaster forces simultaneously all present customers (waiting and served) to abandon the system permanently with a probability p but a weak disaster is that all customers decide to be patient by staying in the system, and wait during the repair time with a probability p, where arrival of a new customer can occur. As soon as the repair process of the server is completed, the server remains providing service in the operational vacation period. We analyze this proposed model and derive the probabilities generating functions of the number of customers present in the system together with explicit expressions of some performance measures such as the mean and the variance of the number of customers in the different states, together with the mean sojourn time. Finally, numerical results are presented to show the influence of the system parameters on some studied performance measures.
... It is noted that IoUT node does not process compression if queue is empty. In the procedure of AQM, it processes packet in the head of the queue one after another, and compresses packet with a probability of or keeps it with a probability of = 1 − , which is modeled as geometric abandonments [11]. Moreover, the AQM stops when the first packet is kept, or all packets are compressed and transmitted. ...
We investigate the peak age of information (PAoI) in underwater wireless sensor networks (UWSNs), where Internet of underwater things (IoUT) nodes transmit the latest packets to the sink node, which is in charge of adjusting the sleep-scheduling to match network demands. In order to reduce PAoI, we propose active queue management (AQM) policy of the IoUT node, beneficially compresses the packets having large waiting time. Moreover, we deduce the closed-form expressions of the average PAoI as well as the energy cost relying on probability generating function and matrix-geometric solutions. Numerical results verify that the IoUT node relying on the AQM policy has a lower PAoI and energy cost in comparison to those using non-AQM policy. Index Terms-Age of information, underwater wireless sensor networks, active queue management, multiple vacation queueing model.
... Both sequential and synchronized abandonments are instigated by distant systems, where customers have to wait for the arrival of an alternative transport provision to leave the system. A detailed study of queueing models with multiple vacations and geometric abandonment may be found in Dimou and Economou (2013); Dimou, Economou, and Fakinos (2011);Panda, Goswami, and Banik (2016) and the references therein. In these studies, waiting customers become impatient when the server is on vacation and decide randomly whether to remain in the system till service initiation or sequentially abandon the system on the availability of a transport facility. ...
This paper studies a single server renewal input queueing model with multiple vacations in the presence of impatient customers. The server follows an exhaustive vacation policy whenever the system is empty. During vacation, waiting customers become impatient and decide independently whether to abandon the system simultaneously or avail service at the vacation completion instant. Using the supplementary variable and difference equation techniques, we obtain the steady-state system-length distributions at pre-arrival and arbitrary epochs. We present the stochastic decomposition structure for the number of customers in the system and derive several performance measures including the sojourn time. Moreover, we examine the impact of various parameters on the system performance measures. A potential application of the model in the inventory management of an e-commerce company with fresh agricultural products is illustrated. Numerical results discuss the effects of the system parameters and provide insight into the performance of such systems.
... Both sequential and synchronized abandonments are instigated by distant systems, where customers have to await for the arrival of an alternative transport provision to abandon the system. A detailed study of queueing models with multiple vacations and geometric abandonment may be found in Dimou et al. (2011); Dimou and Economou (2013); Panda et al. (2016) and the references therein. In these studies, waiting customers become impatient when the server is on vacation and decide randomly whether to remain in the system till service initiation or sequentially abandon the system on the arrival instant of the transport facility. ...
We study a single server renewal input queueing model with multiple vacations and synchronized abandonment. In the absence of the server, customers become impatient and decide independently whether to abandon the system simultaneously or avail service at the vacation completion instants. We use the supplementary variable techniques to find the steady-state system-length distributions at pre-arrival and random epochs. We present the stochastic decomposition for the number of customers and provide several performance measures. Finally, several numerical results are presented to discuss the effects of the system parameters on the system performance.
... In the service protocol, at the arrival times of another homogeneous Poisson process, a random fraction of the queue is selected, removed from the queue, and served simultaneously. Thus, blockchain queueing models Li, Q. L. et al. (2018) can be considered from the viewpoint of the queueing models with catastrophes, as in Chao, X. (1995) and Dimou, S., and Economou, A. (2013). ...
... Yechiali [16] analysed queue with disaster and impatience. Sudesh [13], Dimou and Economou [3] were some of the remarkable papers in queue with disasters and impatience. ...
An / /1 queueing model with disasters and repairs under Bernoulli working vacation schedule is considered. In this model, after every completion of service the server has the choice to choose the normal busy state with probability or he may choose the working vacation state with probability. Also, disasters are allowed to occur in the busy state. In this paper, the stationary PGF of the number of customers in the system and some performance measures are derived.
... It is in general seldom in practical purpose. Dimon and Economon (2013) derived the explicit expressions and computational schemes for various performance descriptors of the single server queue with the catastrophe that occurred according to a Poisson process and types of reneging. Goswami (2014) obtained the steady-state probabilities using displacement operator method for discrete time queuing systems with two heterogeneous servers subject to catastrophe. ...
The concept of catastrophe, happening at random and leading to force to abandon all present customers, data, machines immediately and the instantly inoperative of the service facilities until a new arrival of the customer is not uncommon in many practical problems of the computer and communication systems. In this research article, we present a process to frame the membership function of queuing system' characteristics of the classical single server M/M/1 fuzzy queuing model having fuzzified exponentially distributed inter-arrival time and service time with fuzzy catastrophe. We employ the α-cut approach to transform fuzzy queues into a family of the conventional crisp intervals for the queuing characteristics which are computed with a set of the parametric nonlinear program using their membership functions. We employ basic fuzzy arithmetic fundamental, Zadeh's extension principle, Yager's ranking index to establish fuzzy relation among different rates and to compute corresponding defuzzify values.
... Inverting we get, (12), (16) and (20) in equation (21) gives, Pt is explicitly determined. ...
A single server Markovian queueing model with working vacation subject to disaster and repair is considered. Whenever the server finds nobody in the system, the server is allowed to take a working vacation where, the server provides service at a slower rate than usual. Also disaster can occur either during busy state or during working vacation state. Whenever the system met with disaster all customers are flushed out and the system transits to repair state. Customers are allowed to join the queue even during repair time. After repair, if the server finds customer then the server moves to busy state otherwise the server moves to working vacation state. Using generating function and Laplace transform techniques explicit time dependent probabilities for various states have been obtained.
... Based on the cases of independent and synchronized abandonments, Dimou et al. [4,5] complemented those studies above by considering the case of geometric abandonments. The difference is that the customers decide sequentially whether they will leave the system or not when the abandonment opportunities occur in a vacation. ...
This paper mainly studies customers’ equilibrium balking behavior in Markovian queues with single vacation and geometric abandonments. Whenever the system becomes empty, the server begins a vacation. If it is still empty when the vacation ends, the server stays idle and waits for new arrivals. During a vacation, abandonment opportunities occur according to a Poisson process, and at an abandonment epoch, customers decide sequentially whether they renege and leave the system or not. We consider four information levels: the fully/almost observable cases and the almost/fully unobservable cases, and get the customers’ equilibrium balking strategies, respectively. Then we also get their optimal balking strategies for the almost observable and the almost/fully unobservable cases, and make comparisons of customer strategies and social welfare for the almost observable and the almost/fully unobservable queues with single vacation and multiple vacations. Because of abandonment, we find that the customers’ equilibrium threshold in a vacation may exceed the one in a busy period in the fully observable queues. However, it has little effect on their equilibrium threshold in the almost observable queues, although frequent abandonment opportunity arrival inhibits their optimal threshold. Interestingly, for the almost unobservable queues, customers who arrive in a busy period are not affected by reneging that happened in the previous vacation when they make decisions of joining or balking, whereas the social planner expects that the customers can take it into consideration for social optimization. In the fully unobservable queues, because of no information, possible reneging surely influences customers’ equilibrium and optimal balking behavior. For the almost observable and the almost/fully unobservable queues, the optimal social welfare is greater in the queues with single vacation than that in the queues with multiple vacations.
... It is in general seldom in practical purpose. Dimon and Economon (2013) derived the explicit expressions and computational schemes for various performance descriptors of the single server queue with the catastrophe that occurred according to a Poisson process and types of reneging. Goswami (2014) obtained the steady-state probabilities using displacement operator method for discrete time queuing systems with two heterogeneous servers subject to catastrophe. ...
The concept of catastrophe, happening at random and leading to
force to abandon all present customers, data, machines immediately and the
instantly inoperative of the service facilities until a new arrival of the customer
is not uncommon in many practical problems of the computer and
communication systems. In this research article, we present a process to frame
the membership function of queuing system’ characteristics of the classical
single server M/M/1 fuzzy queuing model having fuzzified exponentially
distributed inter-arrival time and service time with fuzzy catastrophe. We
employ the α-cut approach to transform fuzzy queues into a family of the
conventional crisp intervals for the queuing characteristics which are computed
with a set of the parametric nonlinear program using their membership
functions. We employ basic fuzzy arithmetic fundamental, Zadeh’s extension
principle, Yager’s ranking index to establish fuzzy relation among different
rates and to compute corresponding defuzzify values.
... Dimou et al. (2011) had employed the generating function approach to evaluate the expected number of waited customers in a single vacation queueing model with geometric reneging. Dimou and Economou (2013) applied generating function approach for the single server queues with catastrophes and geometric reneging. ...
In this investigation, we deal with the transient analysis of repairable machining system having multi identical online and warm standby machines. There is facility of repair of failed machines rendered by an unreliable server who is prone to failure and follows the threshold policy in rendering the service and being repaired. The failed machines waiting in the queue may renege and opt sequentially whether to leave the queue or remain in the queue in geometric fashion at the epochs of availability of secondary server. By constructing the transient differential difference equations, the probabilities associated with the system states are determined using the numerical method based on Runge-Kutta method. Various queueing and reliability indices viz. expected number of failed machines in the system, throughput, reliability, mean time to failure, failure frequency, etc., are established and numerically computed by taking illustration. The sensitivity analysis and numerical simulation are performed for exploring the effects of parameters on some performance measures.
... Dimou et al. (2011) had employed the generating function approach to evaluate the expected number of waited customers in a single vacation queueing model with geometric reneging. Dimou and Economou (2013) applied generating function approach for the single server queues with catastrophes and geometric reneging. ...
... Thus, at an abandonment epoch the present customers take decisions all at once and the number in system is reduced following a binomial distribution. A third type of abandonment principle known as geometric or sequential abandonment has been reported in Dimou et al. (2011); Dimou and Economou (2013). This is similar to the synchronized abandonment with the difference that the transport facility has limited capacity, that is, after an individual verification only a few customers that can be well accommodated by the transport facility, leave the system. ...
In this paper, we consider customers’ equilibrium and socially optimal behavior in a single-server Markovian queue with multiple vacations and sequential abandonments. Upon arrival customers decide for themselves whether to join or balk, based on the level of information available to them. During the server’s vacation, present customers become impatient and decide sequentially whether they will abandon the system or not upon the availability of a secondary transport facility. Assuming the linear reward-cost structure, we analyze the equilibrium balking strategies of customers under four cases: fully and almost observable as well as fully and almost unobservable. In all the above cases, the individual and social optimal strategies are derived. Finally, the dependence of performance measures on system parameters are demonstrated via numerical experiments. © 2016 © World Scientific Publishing Co. & Operational Research Society of Singapore
... Sudhesh [21] obtained an explicit transient solution for the state probabilities of the same model studied in [24]. Recently, Dimou and Economou [7] gave a complementing study of [24], where the customers become impatient and leave the system according to a geometric distribution while the server is in repair. Baumann and Sandmann [2] studied a state dependent M/M/c queue with disasters in random environment, and provided a matrix analytic algorithm to obtain the stationary distribution of the queue length. ...
In this paper, we study a single server GI/M/1 queue in a multi-phase service environment with disasters, where the disasters occur only when the server is busy serving customers. Whenever a disaster occurs in an operative service phase, all present customers are forced to leave the system simultaneously, the server abandons the service and an exponential repair time is set on. After the system is repaired, the server resumes his service and moves to service phase i immediately with probability qi,i = 1,2,.,N. Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. After introducing tagged customers and the concept of a cycle, we also derive the sojourn time distribution, the duration of a cycle, and the length of the server's working time in a service cycle. In addition, numerical examples are presented to illustrate the impact of some critical model parameters on performance measures.
Today’s businesses all strive to operate smoothly and efficiently in order to meet customer demandand provide the best possible service because of the intense competition. The queuing theory is crucial in this situation. Queueing models can assist businesses in understanding their performance in advance, enabling themto plan effectively for providing seamless and effective customer service as well as long-term sustainability. Businesses entice customers to sign up for the system by offering promotions and discounts. Customers wait even longer in queue to receive services as a result of incentives like discounts.An analysis is conducted on a finite Markovian single-server queuing model with encouraged arrivals, reneging, and retention of reneged customers. Iterative derivation is used to reach the model's steady state solution. Additionally, the queueing model's performance metrics are gathered. As unique example of this model,several significant queuing models are derived. In order to develop a cost model, a model's economic analysis is presented, and a discussion of numerical representation is also included.Customers are added when they are welcomed, which changes the way theybehave in a queueing system.
Le monde de l’enseignement supérieur a beaucoup changé ces dernières décennies dû à l’omniprésence de l’informatique . Si l’on ajoute à cela les restrictions sanitaires et préventives suite à COVID’19. Les universités rendent les cours en mode distanciel, et dans l’enseignement technique les TP sont très importants et incontournable , et que les laboratoires à distance sont plus bénéfiques par rapport au laboratoire virtuels car ce dernier n’est qu’un modèle informatique d’approximative à la réalité via des modélisations afin de réaliser des simulations diverses. Dans ce contexte, l’EST d’Agadir a développé une plateforme à faible coût appelé LABERSIME installée en cloud (LMS et IDE) dotée d’un système embarqué à base de (ESP32-Micropython) pour piloter des équipements réels de laboratoire et faire des expériences qualitativement efficaces que celles en mode présentiel.
The stability problem of the functional equation
was evoked by Ulam in 1940.
In mathematical modeling of physical problems,
the deviations in measurements will result with
errors and deviations can be dealt with the stability
of equations.
Hence, the stability of equations is essential in
mathematical models.
In this paper, we prove the hyperstability of a cubic
functional equation on a restricted domain.
The method of the proof of the main theorem is
motivated by an idea used by Brzdek in 2013
And further by Piszczek, it is based on a fixed point
theorem for functional spaces obtained by Brzdek
[1-2].
The 4th International Conference on Research in Applied Mathematics and
Computer Science (ICRAMCS 2022) is aimed to bring researchers and
professionals to discuss recent developments in both applied mathematics
and computer science and to create a professional knowledge exchange
platform between mathematicians, computer science and other disciplines.
This conference is the result of international cooperation bringing together
African and European universities. It is a privileged place for meetings and
exchanges between young researchers and high-level African and
international decision makers in the fields of mathematics and applied
computing.
This conference has several major objectives, in particular:
▪ To bring together doctoral students and research professors in the
fields of applied sciences and new technologies.
▪ To consolidate the scientific cooperation between the university and
the socio-economic environment in the field of applied sciences.
▪ To allow young researchers to present and discuss their research
work before a panel of specialists and university professors.
▪ To contribute to the development of a database, which can help
decision makers to opt for a better management strategy.
The abstracts of these conference proceedings were presented at the 4th
International Conference on Research in Applied Mathematics and
Computer Science (ICRAMCS 2022). These conference proceedings include
abstracts that underwent a rigorous review by two or more reviewers.
These papers represent current important work in the field of Mathematics
& Computer Science and are elaborations of the ICRAMCS conference
reports.
This paper studies and compares customer balking behavior in some single-server observable queues with N policies and geometric abandonments. We focus on two types of N policies: the classic one and the modified one. Under the classic N policy, the server begins to take a vacation period as soon as the system becomes empty until there are N customers in the queue. On the other hand, under the modified N policy, the server takes multiple vacations until at least N customers are waiting in the queue when a vacation time ends. Abandonment opportunities occur during a whole vacation period, and customers decide independently and sequentially whether to renege. Comparing the two types of N policies, we realize that the residual time of the server's last vacation under the modified N policy indeed inhibits customers joining, whereas it may be more economic in practice if we take the whole system cost into consideration. As for the optimal N, under which the server is free of compensation, we observe that he should choose the modified N policy when the abandonment probability is much close to 1. Otherwise, he should choose the classic N policy.
The current study considers a Markovian queue, where the server is subject to breakdowns while providing service to customers. During a breakdown period, the server lowers the service rate, rather than completely halting services provision. When there are no customers in the system, the server leaves for a vacation. Upon return from a vacation, if the server finds no customers in the system, the server leaves for another vacation; however, if the server from a vacation to find at least one customer waiting in the queue, the server will serve customers immediately. During a vacation period, the server serves customers at a different service rate and does not stop working. Service times during normal, breakdown and vacation periods follow exponential distributions. For such a system, to find the closed-form solutions for the steady-state probabilities, we employ the spectral expansion method. Several performance measures of the system are developed. We additionally derive the Laplace-Stieltjes transform of the sojourn time of an arbitrary customer and obtain the expected sojourn time. Furthermore, we provide numerical examples to illustrate the effects of various system parameters on performance measures and the expected sojourn time.
This paper investigates an N-policy GI/M/1 queue in a multi-phase service environment with disasters, where the system tends to suffer from disastrous failures while it is in operative service environments, making all present customers leave the system simultaneously and the server stop working completely. As soon as the number of customers in the queue reaches a threshold value, the server resumes its service and moves to the appropriate operative service environment immediately with some probability. We derive the stationary queue length distribution, which is then used for the computation of the Laplace-Stieltjes transform of the sojourn time of an arbitrary customer and the server’s working time in a cycle. In addition, some numerical examples are provided to illustrate the impact of several model parameters on the performance measures.
We consider a stochastic model where a population grows in batches according to renewal arrival process. The population is prone to be affected by catastrophes which occur according to Poisson process. The catastrophe starts the destruction of the population sequentially, with one individual at a time, with probability p. This process comes to an end when the first individual survives or when the entire population is eliminated. Using supplementary variable and difference equation method we obtain explicit expressions of population size distribution in steady-state at pre-arrival and arbitrary epochs, in terms of roots of the associated characteristic equation. Besides, we prove that the distribution at pre-arrival epoch is asymptotically geometric. Based on our theoretical work we present few numerical results to demonstrate the efficiency of our methodology. We also investigate the impact of different parameters on the behavior of the model through some numerical examples.
This paper compares the customers’ balking behavior in site-clearing and non-site-clearing Markovian queues with server failures and unreliable repairer. That is, the system is subject to failures (catastrophes or normal failures) that occur when the server is at a functioning state, and it carries out the site-clearing mechanism if catastrophes occur while it carries out the non-site-clearing mechanism if normal failures occur. At a failure epoch, the server is turned off and all present customers are forced to leave the system in a site-clearing queue, whereas they start to experience a repair process in a non-site-clearing queue. After an exponential repair time, the server can be reactivated only with a probability p . Once the server fails to be repaired, all present customers leave the system, and meanwhile the next repair time begins. Comparing the equilibrium and optimal balking strategies of customers for the two types of queues, we find that although customers more possibly face the bad outcome that can not be served eventually, they still prefer to select a site-clearing queue but not a non-site-clearing queue, which coincides with the social planner’s preference. Moreover, the customers’ equilibrium behavior more deviates from their socially optimal behavior in the site-clearing queues.
This study deals with the performance modeling and reliability analysis of a redundant machining system composed of several functional machines. To analyze the more realistic scenarios, the concepts of switching failure and geometric reneging are included. The time-to-breakdown and repair time of operating and standby machines are assumed to follow the exponential distribution. For the quantitative assessment of the machine interference problem, various performance measures such as meantime to failure , reliability, reneging rate, etc. have been formulated. To show the practicability of the developed model, a numerical illustration has been presented. For the practical justification and validity of the results established, the sensitivity analysis of reliability indices has been presented by varying different system descriptors. Growing Science Ltd. All rights reserved. 7
This study deals with the performance modeling and reliability analysis of a redundant machining system composed of several functional machines. To analyze the more realistic scenarios, the concepts of switching failure and geometric reneging are included. The time-to-breakdown and repair time of operating and standby machines are assumed to follow the exponential distribution. For the quantitative assessment of the machine interference problem, various performance measures such as meantime to failure , reliability, reneging rate, etc. have been formulated. To show the practicability of the developed model, a numerical illustration has been presented. For the practical justification and validity of the results established, the sensitivity analysis of reliability indices has been presented by varying different system descriptors. Growing Science Ltd. All rights reserved. 7
This study deals with the performance modeling and reliability analysis of a redundant machining system composed of several functional machines. To analyze the more realistic scenarios, the concepts of switching failure and geometric reneging are included. The time-to-breakdown and repair time of operating and standby machines are assumed to follow the exponential distribution. For the quantitative assessment of the machine interference problem, various performance measures such as meantime to failure , reliability, reneging rate, etc. have been formulated. To show the practicability of the developed model, a numerical illustration has been presented. For the practical justification and validity of the results established, the sensitivity analysis of reliability indices has been presented by varying different system descriptors. Growing Science Ltd. All rights reserved. 7
A queueing model with catastrophes and delayed action is studied in this paper. This delayed action could be in the form of protecting or removing all the customers that are in the system based on the outcome of two random clocks which are simultaneously activated upon the occurrence of a catastrophic event. Assuming the customers to arrive according to a versatile Markovian point process to a single server system, the service times to be of phase type, and all other underlying random variables to be exponentially distributed, we use matrix-analytic methods to study the delayed catastrophic model in steady-state. Needed expressions for the number in the system as well as the waiting time distributions are derived along with a discussion on some special cases of this model. Detailed illustrative examples are presented.
In this paper we present various approaches for the transient analysis of a Markovian population process with total catastrophes. We discuss the pros and the cons of these methodologies and point out how they lead to different tractable extensions. As an illustrating example, we consider the non-homogeneous Poisson process with total catastrophes. The extension of the probabilistic methodologies for analyzing models with binomial catastrophes is also discussed.
For the M/M/1 queue in the presence of catastrophes the transition probabilities, densities of the busy period and of the catastrophe waiting time are determined. A heavy-traffic approximation to this discrete model is then derived. This is seen to be equivalent to a Wiener process subject to randomly occurring jumps for which some analytical results are obtained. The goodness of the approximation is discussed by comparing the closed-form solutions obtained for the continuous process with those obtained for the M/M/1 catastrophized queue.
We present a simple algorithm for numerically inverting Laplace transforms. The algorithm is designed especially for probability cumulative distribution functions, but it applies to other functions as well. Since it does not seem possible to provide effective methods with simple general error bounds, we simultaneously use two different methods to confirm the accuracy. Both methods are variants of the Fourier-series method. The first, building on Dubner and Abate (Dubner, H., J. Abate. 1968. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. JACM 15 115–123.) and Simon, Stroot, and Weiss (Simon, R. M., M. T. Stroot, G. H. Weiss. 1972. Numerical inversion of Laplace transforms with application to percentage labeled experiments. Comput. Biomed. Res. 6 596–607.), uses the Bromwich integral, the Poisson summation formula and Euler summation; the second, building on Jagerman (Jagerman, D. L. 1978. An inversion technique for the Laplace transform with applications. Bell System Tech. J. 57 669–710 and Jagerman, D. L. 1982. An inversion technique for the Laplace transform. Bell System Tech. J. 61 1995–2002.), uses the Post-Widder formula, the Poisson summation formula, and the Stehfest (Stehfest, H. 1970. Algorithm 368. Numerical inversion of Laplace transforms. Comm. ACM 13 479–490 (erratum: 13 624).) enhancement. The resulting program is short and the computational experience is encouraging.
INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.
A transient solution is obtained analytically using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and non-zero repair time. The steady state probability of the system
size is present. Some key performance measures, namely, throughput, loss probability and response time for the system under
consideration are investigated. Further, reliability and availability of the system are analysed. Finally, numerical illustrations
are used to discuss the system performance measures.
Consider a system (e.g. a computer farm or a call center) operating as a M/M/c queue, where c = 1, or 1 < c < 1, or c = 1. The system as a whole suffers disastrous breakdowns, resulting in the loss of all running and waiting sessions. When the system is down and undergoing a repair process, newly arriving customers become impatient: each individual customer activates a random-duration timer. If the timer expires before the system is repaired, the customer abandons the queue. We analyze this model and derive various quality of service measures: mean sojourn time of a served customer; proportion of customers served; rate of lost customers due to disasters; and rate of abandonments due to impatience.
In this paper we present a detailed analysis of queueing models with vacations and impatient customers, where the source of impatience is the absence of the server. Instead of the standard assumption that customers perform independent abandonments, we consider situations where customers abandon the system simultaneously. This is, for example, the case in remote systems where customers may decide to abandon the system, when a transport facility becomes available.
Consider a system performing a continuous-time random walk on the integers,
subject to catastrophes occurring at constant rate, and followed by
exponentially-distributed repair times. After any repair the system starts anew
from state zero. We study both the transient and steady-state probability laws
of the stochastic process that describes the state of the system. We then
derive a heavy-traffic approximation to the model that yields a jump-diffusion
process. The latter is equivalent to a Wiener process subject to randomly
occurring jumps, whose probability law is obtained. The goodness of the
approximation is finally discussed.
The growth of populations with continuous deterministic and random jump components is treated. Three special models in which random jumps occur at the time of events of a Poisson process and admit formal explicit solutions are considered: A) Logistic growth with random disasters having exponentially distributed amplitudes; B) Logistic growth with random disasters causing the removal of a uniformly distributed fraction of the population size; and C) Exponential decay with sudden increases (bonanzas) in the population and with each increase being an exponentially distributed fraction of the current population. Asymptotic and numerical methods are employed to determine the mean extinction time for the population, qualitatively and quantitatively. For Model A, this time becomes exponentially large as the carrying capacity becomes much larger than the mean disaster size. Implications for colonizing species for Model A are discussed. For Model B and C, the practical notion of a small, but positive, effective extinction level is chosen, and in these cases the expected extinction time rises rapidly with population size, yet at less than an exponentially large order.
The second edition has greatly benefited from a sizable number of comments and suggestions I received from users of the book. I hope that I have corrected all the er rors and misprints in the book. Important revisions were made in Chapters I and 4. In Chapter I, we added two appendices (global stability and periodic solutions). In Chapter 4, we added a section on applications to mathematical biology. Influenced by a friendly and some not so friendly comments about Chapter 8 (previously Chapter 7: Asymptotic Behavior of Difference Equations), I rewrote the chapter with additional material on Birkhoff's theory. Also, due to popular demand, a new chapter (Chapter 9) under the title "Applications to Continued Fractions and Orthogonal Polynomials" has been added. This chapter gives a rather thorough presentation of continued fractions and orthogonal polynomials and their intimate connection to second-order difference equations. Chapter 8 (Oscillation Theory) has now become Chapter 7. Accordingly, the new revised suggestions for using the text are as follows. The diagram on p. viii shows the interdependence of the chapters The book may be used with considerable flexibility. For a one-semester course, one may choose one of the following options: (i) If you want a course that emphasizes stability and control, then you may select Chapters I, 2, 3, and parts of 4, 5, and 6. This is perhaps appropriate for a class populated by mathematics, physics, and engineering majors.
A particular random walk on the integers leads to a new, tractable Markov chain. Of the stationary probabilities, we discuss the existence, some analytic properties and a factorization which leads to an algorithmic procedure for their numerical computation. We also consider the positive recurrence of some variants which each call for different mathematical arguments. Analogous results are derived for a continuous version on the positive reals.
A particular random walk on the integers leads to a new, tractable Markov chain. Of the stationary probabilities, we discuss the existence, some analytic properties and a factorization which leads to an algorithmic procedure for their numerical computation. We also consider the positive recurrence of some variants which each call for different mathematical arguments. Analogous results are derived for a continuous version on the positive reals.
We consider Markov models for growth of populations subject to catastrophes. Emphasis is placed on discrete-state models where immigration is possible and the catastrophe rate is population-dependent. Explicit formulas for descriptive quantities of interest are derived when catastrophes reduce population size by a random amount which is either geometrically, binomially or uniformly distributed. Comparison is made with continuous-state Markov models in the literature in which population size evolves continuously and deterministically upwards between random jumps downward.
The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfy for each , and Σ1 ∞ λj -1 = ∞. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj = jλ, μj = jμ) with catastrophes of several different types.
This paper explores an alternative approach starting from first principles, to the derivation of probability generating functions (pgfs) of death, birth-death and immigration processes in continuous time, subject to random catastrophes. A more elementary version of the general method proposed by Economou and Fakinos (2003) is presented. We examine the simple death process, the survival of susceptibles in a carrier-borne epidemic, the birth-death and immigration process, the unbiased random walk and the barber shop queue, all of them subject to random catastrophes occurring as a Poisson process. The stationary pgfs and the expected values of the processes are derived.
Several recent papers have investigated the customer reneging behavior in queueing systems with vacations, where the customers become impatient during the absence of the server(s). These studies have treated the cases of independent and synchronized abandonments. In the case of independent abandonments, the customers have their own independent patience times and abandon the system when such times expire. In the case of synchronized abandonments, the abandonment opportunities occur according to a certain point process and then all present customers decide simultaneously but independently whether they will abandon the system or not.In the present paper, we complement these studies by considering the case of geometric abandonments. This case arises when the abandonment opportunities occur according to a certain point process and the customers decide sequentially whether they will leave the system or not. We derive explicit expressions and computational schemes for various performance descriptors, including the number of customers in system, the sojourn time of a customer, the duration and the maximum number of customers in a busy period.
A transient solution for the system size in the M/M/1 queueing model with the possibility of catastrophes at the service station is derived in the direct way. Asymptotic behavior of the probability of the server being idle and mean queue size are discussed. Steady-state probabilities are also obtained.
We deal with a population of individuals that grows stochastically according to a batch Markovian arrival process and is subject to renewal generated geometric catastrophes. Our interest is in the semi-regenerative process that describes the population size at arbitrary times. The main feature of the underlying Markov renewal process is the block structure of its embedded Markov chain. Specifically, the embedded Markov chain at post-catastrophe epochs may be thought of as a Markov chain of GI/G/1-type, which is indeed amenable to be studied through its R- and G-measures, and a suitably defined Markov chain of M/G/1-type. We present tractable formulae for a variety of probabilistic descriptors of the population, including the equilibrium distribution of the population size and the distribution of the time to extinction for present units at post-catastrophe epochs.
In this chapter, our interest is in determining the stationary distribution of an irreducible positive recurrent Markov chain with an infinite state space. In particular, we consider the solution of such chains using roots or zeros. A root of an equation f (z) = 0 is a zero of the function f (z),and so for notational convenience we use the terms root and zero interchangeably. A natural class of chains that can be solved using roots are those with a transition matrix that has an almost Toeplitz structure. Specifically, the classes of M/G/1 type chains and G/M/1 type chains lend themselves to solution methods that utilize roots. In the M/G/1 case, it is natural to transform the stationary equations and solve for the stationary distribution using generating functions. However, in the G/M/1 case the stationary probability vector itself is given directly in terms of roots or zeros. Although our focus in this chapter is on the discrete-time case, we will show how the continuous-time case can be handled by the same techniques. The M/G/1 and G/M/1 classes can be solved using the matrix analytic method [Neuts, 1981, Neuts, 1989], and we will also discuss the relationship between the approach using roots and this method.
We study the asymptotic behavior of maximum values of birth and death processes over large time intervals. In most cases, the distributions of these maxima, under standard linear normalizations, either do not converge or they converge to a degenerate distribution. However, by allowing the birth and death rates to vary in a certain manner as the time interval increases, we show that the maxima do indeed have three possible limit distributions. Two of these are classical extreme value distributions and the third one is a new distribution. This third distribution is the best one for practical applications. Our results are for transient as well as recurrent birth and death processes and related queues. For transient processes, the focus is on the maxima conditioned that they are finite.
Our note1 is dedicated to the Palm/Erlang-A Queue. This is the simplest practice- worthy queueing model, that accounts for customers' impatience while waiting. The model is gaining importance in support of the stang of call centers, which is a central step in their Service-Engineering. We discuss computations of performance measures, both theoretical and software-based (via the 4CallCenter software). Then several examples of Palm/Erlang- A applications are presented, mostly motivated by and based on real call center data.
Markov branching processes and in particular birth-and-death processes are considered under the influence of disasters that arrive independently of the present population size. For these processes we derive an integral equation involving a shifted and rescaled argument. The main emphasis, however, is on the (random) probability of extinction. Its distribution density satisfies an equation which can be solved numerically at least up to a multiplicative constant. In an example it is also found by simulation.
We consider a continuous-time Markov chain with state space S. A point process {X(t),t⩾0}, using a function A:[0,∞)→S in the following way, influences the evolution of : whenever an event of {X(t)} occurs at time t, the Markov chain of interest goes immediately at state A(t) and evolves according to the dynamics of until the next event of {X(t)} and so on. We study the transient and the limiting distribution of the resulting process and prove some stochastic comparison facts. Several examples that demonstrate the applicability of the model are also included.
We consider a single server unreliable queue represented by a 2-dimensional continuous-time Markov chain. At failure times, the present customers leave the system. Moreover, customers become impatient and perform synchronized abandonments, as long as the server is down. We analyze this model and derive the main performance measures using results from the basic q-hypergeometric series.
A new approach is used to determine the transient probability functions of the classical queueing systems: M/M/1, M/M/1/H, and M/M/1/H with catastrophes. This new solution method uses dual processes, randomization and lattice path combinatorics. The method reveals that the transient probability functions for M/M/1/H and M/M/1/H with catastrophes have the same mathematical form.
This paper deals with the analysis of an M/M/c queueing system with setup times. This queueing model captures the major characteristics of phenomena occurring in production
when the system consists in a set of machines monitored by a single operator. We carry out an extensive analysis of the system
including limiting distribution of the system state, waiting time analysis, busy period and maximum queue length.
This is an introductory-level text on stochastic modeling. It is suited for undergraduate students in engineering, operations research, statistics, mathematics, actuarial science, business management, computer science, and public policy. It employs a large number of examples to teach the students to use stochastic models of real-life systems to predict their performance and use this analysis to design better systems.
In this note a simple immigration--birth--death process is considered, which is influenced by total catastrophes that are introduced with constant rate. The stationary probabilities are derived using a renewal argument.
For a modified Anderson and May model of host parasite dynamics it is shown that infections of different levels of virulence die out asymptotically except those that optimize the basic reproductive rate of the causative parasite. The result holds under the assumption that infection with one strain of parasite precludes additional infections with other strains. Technically, the model includes an environmental carrying capacity for the host. A threshold condition is derived which decides whether or not the parasites persist in the host population.
Under the influence of randomly occurring disasters, the eventual extinction probability, q, of a birth and death process, Z, is a random variable. In this paper, we obtain an integral expression for the probability density function g(x) of q under the assumption that the population process Z is a time homogeneous linear birth and death process and the disasters occur according to an arbitrary renewal process so that its interarrival times have a density. An example is provided to demonstrate how to evaluate the integral numerically.
Populations are often subject to the effect of catastrophic events that cause mass removal. In particular, metapopulation models, epidemics, and migratory flows provide practical examples of populations subject to dis asters (e.g., habitat destruction, environmental catastrophes). Many stochastic models have been developed to explain the behavior of these populations. Most of the reported results concern the measures of the risk of extinction and the distribution of the population size in the case of total catastrophes where all individuals in the population are removed simultaneously. In this paper, we investigate the basic immigration process subject to binomial and geometric catastrophes; that is, the population size is reduced according to a binomial or a geometric law. We carry out an extensive analysis including first extinction time, number of individuals removed, survival time of a tagged individual, and maximum population size reached between two consecutive extinctions. Many explicit expressions are derived for these system descriptors, and some emphasis is put to show that some of them deserve extra attention.
Transient analysis of a single server queue with catastrophes, failures and repairs Modeling and analysis of stochastic systems Stationary probabilities for a simple immigration-birth-death process under the influence of total catastrophes
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R, Buhler WJ, Chan W, Pearl DK (1989) Population processes under the influence of disasters occurring independently of population size. J Math Biol 27:179–190 Brockwell PJ (1986) The extinction time of a general birth and death process with catastrophes. J Appl Probab 23:851–858 Brockwell PJ, Gani J, Resnick SI (1982) Birth, immigration and catastrophe processes. Adv Appl Probab 14:709–731
Synchronized abandonments in a single server unreliable queue An introduction to difference equations Use of charasteristic roots for solving infinite state Markov chains
- A Kapodistria
A, Kapodistria S (2010) Synchronized abandonments in a single server unreliable queue. Eur J Oper Res 203:143–155 Elaydi SN (1999) An introduction to difference equations, 2nd edn. Springer-Verlag, New York Gail HR, Hantler SL, Taylor BA (2000) Use of charasteristic roots for solving infinite state Markov chains. In: Grassmann WK (ed) Computational probability. Kluwer, pp 205–255
The distribution of the maximum orbit size of an M/G/1 retrial queue during the busy period Advances in stochastic modelling
- Mj López-Herrero
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+ G queue: summary of performance measures
- A Mandelbaum
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English translation of results first published in 1946 in Swedish in the same journal, which was then entitled
- C Palm
The distribution of the maximum orbit size of an M/G/1 retrial queue during the busy period
- M J López-Herrero
- M F Neuts
- MJ López-Herrero
The M/M/n + G queue: summary of performance measures
- A Mandelbaum
- S Zeltyn