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Analysis of a Non-Preemptive Priority Multiserver Queue
Abstract
We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.
... The extent to which the research works described above can be applied to priority service systems is however limited, due to the extra level of complexity associated with prioritization rules. For this reason, the majority of research papers analyzing time-dependent priority queues have tended to focus on the long-run steady-state performance of the system (Gail, Hantler, & Taylor, 1988; Kao and Wilson, 1999). Although a novel matrix-analytic method to analyze the expected waiting time of two customer classes in multiple priority dual queues has recently been proposed by Zeephongsekul and Bedford (2006), their analysis is again restricted to scenarios where there is a single server and a consistent arrival rate. ...
... Although a novel matrix-analytic method to analyze the expected waiting time of two customer classes in multiple priority dual queues has recently been proposed by Zeephongsekul and Bedford (2006), their analysis is again restricted to scenarios where there is a single server and a consistent arrival rate. The research contained within this article extends the methodology of Gail et al. (1988) to cover priority queueing systems and fuses it with the MDCTMC approach introduced by Ingolfsson (2005) for M(t)/M/s(t) queues, in order to provide a tractable approach to track behavior in time-dependent priority queues. We importantly further define the corresponding instantaneous transitions necessary to correct for situations where (i) the entire workforce turns over at truly exhaustive " full " shift boundaries, and (ii) only minor changes are made to the workforce, at instants we coin " partial " shift boundaries (i.e., instants where small adjustments are made to the staffing function to more closely align capacity with peaks and troughs in demand). ...
... In order to accurately track the movement of all customers through the system, it is necessary to compute the number of customers of types i, j, h, and l in the system over time, represented by the quadruple S = (i, j, h, l). Following the methodology presented by Gail et al. (1988), it is easily shown that the description of this state space quadruple S = (i, j, h, l) may be reduced to r S = (i, j ) if at least one server is idle (as both h and l must both be null, because there will be no customers in the queue) r S = (i, h, l) if all servers are busy (because j may be derived from the description of the other parameter values) As such, this convenient notation simplifies the state space description and increases the computational efficiency of a numerical solver to track system behavior over time. The equilibrium equations that define the evolution of the system are well known for M/M/s/FIFO queues (Gross & Harris, 1998). ...
This article addresses the optimal staffing problem for a nonpreemptive priority queue with two customer classes and a time-dependent arrival rate. The problem is related to several important service settings such as call centers and emergency departments where the customers are grouped into two classes of “high priority” and “low priority,” and the services are typically evaluated according to the proportion of customers who are responded to within targeted response times. To date, only approximation methods have been explored to generate staffing requirements for time-dependent dual-class services, but we propose a tractable numerical approach to evaluate system behavior and generate safe minimum staffing levels using mixed discrete-continuous time Markov chains (MDCTMCs). Our approach is delicate in that it accounts for the behavior of the system under a number of different rules that may be imposed on staff if they are busy when due to leave and involves explicitly calculating delay distributions for two customer classes. Ultimately, we embed our methodology in a proposed extension of the Euler method, coined Euler Pri, that can cope with two customer classes, and use it to recommend staffing levels for the Welsh Ambulance Service Trust (WAST).
... Let us review some related papers. Gail, Hantler, and Taylor [5] studied the nonpreemptive multiserver version of the M/M/c queue with two classes of jobs. Kao and Narayanan [8] used the matrix-geometric approach in conjunction with the state reduction method to develop a computationally efficient procedure for solving the model presented by Gail et al. [5]. ...
... Gail, Hantler, and Taylor [5] studied the nonpreemptive multiserver version of the M/M/c queue with two classes of jobs. Kao and Narayanan [8] used the matrix-geometric approach in conjunction with the state reduction method to develop a computationally efficient procedure for solving the model presented by Gail et al. [5]. Wagner [19] recently studied the nonpreemptive multiserver system with k classes of jobs, where k could be any positive finite integer. ...
... He allows a MAP arrival process and phase-type processing times, with the same dimension of phases for all job types. These papers [5, 8, 19] did not present any procedures for obtaining the waiting time distributions. Takine [16, 17] studied the continuoustime nonpreemptive priority MAP/G/1 queue, and derived various formulas on the generating function of the queue length distribution and the Laplace–Stieltjes transform (LST) of the waiting time for each class of customers. ...
We use the matrix-geometric method to study the MAP/PH/1 general preemptive priority queue with a multiple class of jobs. A procedure for obtaining the block matrices representing the transition matrix P is presented. We show that the special upper triangular structure of the matrix R obtained by Miller (Computation of steady-state probabilities for M/M/1 priority queues, Oper Res 29(5) (1981), 945-958) can be extended to an upper triangular block structure. Moreover, the subblock matrices of matrix R also have such a structure. With this special structure, we develop a procedure to compute the matrix R. After obtaining the stationary distribution of the system, we study two primary performance indices, namely, the distributions of the number of jobs of each type in the system and their waiting times. Although most of our analysis is carried out for the case of K 3, the developed approach is general enough to study the other cases (K 4). © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 662- 682, 2003.
... For larger systems (eight servers) they begin to experience numerical stability problems – their solution yields some negative probabilities . Gail, Hantler, and Taylor [7, 8] follow a similar approach and also report stability problems. Feng, Kowada, and Adachi [6] generalize priority service to more general switching thresholds. ...
... Furthermore our method allows general PH job sizes and is highly accurate; see validation Section 5. By contrast , the prior literature on the M/M/¡ priority queue (exponential job sizes) has sometimes resulted in numerically unstable compu- tations [7, 8] or complicated partitions of the state space [18, 23]. ...
We ask the question, "for minimizing mean response time, which is preferable: one fast server of speed , or slow servers each of speed ?" Our setting is the M/GI/ system with two priority classes of customers, high priority and low priority, where is a phase-type distribution. We find that multiple slow servers are often preferable — and we demonstrate exactly how many servers are preferable as a function of load and . In addition, we find that the optimal number of servers with respect to the high priority jobs may be very different from that preferred by low priority jobs, and we characterize these preferences. We also evaluate the optimal number of servers with respect to overall mean response time, averaged over high and low priority jobs. Lastly, we ascertain the effect of the variability of high priority jobs on low priority jobs. This paper is the first to analyze an M/GI/ system with two priority classes and a general phase-type distribution. Prior analyses of the M/GI/ with two priority classes either require that be exponential, or are approximations that work well when is exponential, but are less reliable for more variable . Our analytical method is very different from the prior literature: it combines the technique of dimensionality reduction (see (9)) with Neuts' technique for determining busy periods in multiserver systems (22). Our analysis is approximate, but can be made as accurate as desired, and is verified via simulation.
... There are a number of approximations based on aggregation or truncation of the state space [11, 12, 23, 14], sometimes in combination with the matrix analytic method and state space partitioning [22]. All the above papers assume preemptive-resume priorities (as in this paper), but there also are a few papers on non-preemptive priorities [8, 13, 29]. Exact analyses for two priority classes with exponential job size distributions either use (i) matrix analytic methods or (ii) generating function methods. ...
... shown inFigure 1, and now differentiate between 0, 1, 2, or 3-or-more high priority jobs. This can be easily extended to the case of ¤ By contrast, the prior literature on the M/M/ ¤ priority queue has sometimes resulted in numerically unstable computations [8, 9] or complicated partitions of the state space [19, 22]. ...
Computer systems depend on high priority background processes to provide both reliability and security. This is especially true in multiserver systems where many such background processes are required for data coherence, fault detection, intrusion detection, etc. From a user's perspective, it is important to understand the effect that these many classes of high priority, background tasks have on the performance of lower priority user-level tasks. We model this situation as an M/GI/ queue with preemptive-resume priority classes, presenting the first analysis of this system with more than two priority classes under a general phase-type service distribution. (Prior analyses of the M/GI/ with more than two priority classes are approximations that, we show, can be highly inaccurate.) Our analytical method is very different from the prior literature: it combines the tech- nique of dimensionality reduction (10) with Neuts' technique for determining busy periods in multiserver systems (21), and then uses a novel recursive iteration technique. Our analysis is approximate, but, unlike prior techniques, can be made as accurate as desired, and is verified via simulation.
... Constraints (19)-(22) are the queuing equations that control the stability of charging servers. Based on Little (1961) and Gail et al. (1988), waiting time is a function of arrival rate and service rate. Therefore changing the arrival rate according to the service level of each station can lower the waiting time at the station as well. ...
... Sufficient conditions for ergodicity and transience of the Markov process can be obtained by using criteria based on a Lyapunov function or test function. For the stability analysis of queueing models, ergodicity and nonergodicity criteria with test functions have been used, see, for example, [4,8,10,11,13]. ...
We consider a queueing system with two classes of customers, two heterogeneous servers, and discriminatory random order service (DROS) dis cipline. The two servers may have either the same or different DROS weights for each class. Customers of each class arrive according to a Poisson process and the service times of each class of customers are assumed to be exponen tially distributed with service rate depending on both the customer's class and the servers. We provide stability and instability conditions for this two-class two-server queue with DROS discipline.
... There is also a number of papers studying the joint queue length distribution using alternative approaches. Generating functions are used in [9, 10] for the analysis of M/M/c priority queueing systems with two classes. Generating functions are also used in [19] for an M/M/c preemptive priority system with more than two classes. ...
... Sapna Isotupa and Stanford (2002) have extended Miller's work to the ∑i = 1NMi/PHi/1 non-preemptive priority queues. The non-preemptive multi server Markov model with two classes of customers was considered by Kao and Narayanan (1990), and Gail et al. (1988). The non-preemptive priority model with MAP arrivals, two-classes of customers and general distributed service times was by proposed by Takine (1996). ...
Spare support plays an important role in improving the reliability of multi-component repairable machining systems in many industries as well as in day-to-day real-time embedded systems. This paper is concerned with the reliability analysis of embedded machining system consisting of two types of units along with warm and cold standbys support under the priority concepts. The Markov model is developed by constructing the transient equations using birth death process. Various queueing and reliability performance measures are established in terms of transient probabilities of the system states which are evaluated using numerical technique based on Runge-Kutta method. To illustrate the tractability of the proposed method, a numerical example is worked out. Further neuro-fuzzy inference approach is employed to compare some performance indices which are also obtained numerically.
... There are so many approaches to calculate the queue length and waiting time for each class. Gail, Kao, Wagner et al have studied multiserver non-preemptive model with two priority classes [17] [18] [19]. Kao implemented a power–series method for the two priority queues. ...
Recently Television White Spaces are used without license according to FCC rule. It is based on dynamically selection of available TV spectrum (white spaces). It gives a possibility of new sets of applications. In vehicular wireless communication system, these white spaces can easily be used on an opportunistic base. To model a multi-access multiuser architecture and analyze network performances in both wired and wireless system queueing theory has been massively used. In this paper we modeled a vehicular dynamic spectrum access (VDSA) system by using queueing theory which uses vacant TV channels for communication. Dynamically assigned allowed available channels are modeled as servers. To efficient use of these limited number of channels we defined some non-preemptive priority classes among users. The availability of servers is occurred in a time-location phenomenon. This paper examined the feasibility analysis of vehicular dynamic spectrum access performance through TV white spaces via multi server multi priority non-preemptive queueing theory. Both M/M/m and M/G/m models are employed to evaluate the probability of all channels are busy, the amount of channel utilizations by different priority class users and also to calculate the transmission latencies.
... Since then, the priority queueing systems have attracted a lot of interest. See, for example, Gail, Hantler and Taylor [7] [8], Kao and Narayanan [11], Takine [17], Alfa [1], Isotupa and Stanford [10], Alfa, Liu and He [2], Drekic and Woolford [4], Zhao et al. [19] and so on. The stationary distribution is very important for a queueing model to characterize its performance, but in most cases, it is difficult to obtain the explicit expression of the stationary distribution. ...
In this paper, we consider a discrete-time preemptive priority queue with
different service rates for two classes of customers, one with high-priority
and the other with low-priority. This model corresponds to the classical
preemptive priority queueing system with two classes of independent Poisson
customers and a single exponential server. Due to the possibility of customers'
arriving and departing at the same time in a discrete-time queue, the model
considered in this paper is more complicated. In this model, we focus on the
characterization of exact tail asymptotics for the joint stationary
distribution of the queue length of the two classes of customers, for the two
boundary distributions and for the two marginal distributions, respectively. By
using generating functions and kernel method, we get an explicit expression of
exact tail asymptotics along the low-priority queue direction, as well as along
the high-priority queue direction.
... If the service times are different, they derive an approximation, based on replacing k servers by a single server that works k times as fast. Mitrany and King [13] and Gail et al. [14,15] derive a method to obtain the equilibrium probabilities for a model with two priority classes. None of the authors considers priority classes consisting of customer subclasses having different service times. ...
We consider a multi-server queuing model with two priority classes that consist of multiple customer types. The customers belonging to one priority class customers are lost if they cannot be served immediately upon arrival. Each customer type has its own Poisson arrival and exponential service rate. We derive an exact method to calculate the steady state probabilities for both preemptive and nonpreemptive priority disciplines. Based on these probabilities, we can derive exact expressions for a wide range of relevant performance characteristics for each customer type, such as the moments of the number of customers in the queue and in the system, the expected postponement time and the blocking probability. We illustrate our method with some numerical examples.
Problem definition: Many service and manufacturing systems use both capacity rationing (CR) and priority to differentiate among their customers. We model these as a two-class nonpreemptive priority [Formula: see text] queueing model and the practice of CR; an arriving low-priority customer can directly enter service only when the number of idle servers is higher than the CR level, k. For these systems, we separately discuss two important features that are common in practice but ignored in the literature; supply is narrowly matched with demand, and service rates are heterogeneous, reflecting different customer types. Methodology and results: When the service times of both classes are identical, our asymptotic results indicate that for a system with a large number of servers, the nondegenerative CR level does not exceed [Formula: see text]. When the service times of classes differ, we derive exact solutions for different performance measures of interest using queueing and Markov chain decomposition. We numerically demonstrate the impact of system parameters on these performance measures and provide insights on the CR level. Management implications: We show that as predicted by the asymptotic analysis, an [Formula: see text] CR level can significantly reduce the waits of high-priority customers with little effect on low-priority customers’ waiting. We establish that this insight is robust to heterogeneous service times across classes and other system parameters, such as the number of servers and the arrival rates of the classes.
Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.0106 .
We study service systems with parallel servers and random customer arrivals and focus on the waiting cost of customers. Using a Markov decision process (MDP) modeling approach, we analytically characterize the structures of the optimal dynamic server assignment policies for two important systems, one consisting of multiple homogeneous servers and two classes of customers and the other consisting of two heterogeneous servers and multiple classes of customers. Based on the obtained results, we propose a threshold-type heuristic policy for the generalized system consisting of multiple heterogeneous servers and multiple classes of customers. To design such a heuristic policy, we first develop techniques for the performance evaluation of general threshold-type policies with any given threshold values. We then construct a path to search for the optimal threshold values. We compare the performance of the best threshold-type heuristic policy with that of the optimal policy and show that our proposed heuristic policy is computationally efficient yet generates great performance. To derive additional managerial insights, we compare the system under our threshold-type dynamic server assignment policy with other commonly seen and simple systems, such as the dedicated system and the work-conserving flexible priority system. The clear performance advantage observed from extensive numerical experiments demonstrates the importance and usefulness of dynamic server assignment control for systems serving multiple classes of customer arrivals. Finally, we extend our analysis to incorporate customer-dependent service rates and sojourn-time minimization performance metrics.
This paper was accepted by Chung Piaw Teo, optimization.
Funding: This work was supported by the Hong Kong Research Grants Council [16500921, 16505819] and the National Natural Science Foundation of China [72201233, 72301222].
Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01228 .
Explicit results are derived using simple and exact methods for the joint and marginal queue-length distributions for the M/M/c queue with two non-preemptive priority levels. Equal service rates are assumed. Two approaches are considered. One is based on numerically robust quadratic recurrence relations. The other is based on a complex contour-integral representation that yields exact closed-form analytical expressions, not hitherto available in the literature, that can also be evaluated numerically with very high accuracy.
A new approach is developed for the joint queue-length distribution of the two-level non-pre-emptive M/M/c (i.e. Markovian) priority queue that allows explicit and exact results to be obtained. Marginal distributions are derived for the general multi-level problem. The results are based on a representation of the joint queue-length probability mass function as a single-variable complex contour integral, which reduces to a real integral on a finite interval arising from a cut on the real axis. Both numerical quadrature rules and exact finite sums, involving Legendre polynomials and their generalization, are presented for the joint and marginal distributions. A high level of accuracy is demonstrated across the entire ergodic region. Relationships are established with the waiting-time distributions. Asymptotic behaviour in the large queue-length regime is extracted.
Facing intensified interport competition in the global container shipping market, an increasing number of ports choose to offer berthing priority for carriers to increase their attractiveness. This study is the first to theoretically analyze the efficiency impacts of such prioritization. Specifically, this study models the steady-state dynamics for each terminal in a biterminal port as a prioritized queuing system. We explore the equilibrated shipping flow distribution and resulting total system cost (i.e., bunker consumption cost and waiting time cost) with and without priority provision, along with their major analytical properties. Then, we examine the “second-order” effects of these priority schemes on just-in-time (JIT) arrivals, an increasingly popular green port management tool. Specifically, we investigate how the equilibrium state associated with JIT arrivals could change with priority berthing. These analyses generate some interesting results, including (1) the total system cost increases or remains unchanged when a priority scheme is implemented under a symmetric port with equal service capacities for both terminals; (2) under the asymmetric biterminal case, however, it is also possible that berth prioritization could reduce the total system cost, and such phenomenon occurs only if the terminal which offers prioritization owns larger service capacity; (3) the results indicate that the “price of prioritization” could reach [Formula: see text] in port operation when the berth loading is heavy, implying that priority provision may significantly harm the operational efficiency; and (4) lastly, priority provision has a negative second-order effect on JIT strategies in a symmetric port, and such negative effect may neutralize the positive ones. Those theoretical results are validated by numerical experiments, and some of them are also supported by empirical data. The results provide important practical implications for the decision making of the port (or terminal) agencies.
Funding: This work was supported by the National Natural Science Foundation of China [Grants 72371143 and 72188101].
Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0411 .
We consider a non-preemptive multiserver queue with multiple priority classes. We assume distinct exponentially distributed service times and separate quasi-Poisson arrival processes with a predefined maximum number of requests that can be present in the system for each class. We present an approximation method to obtain the steady-state probabilities for the number of requests of each class in our system. In our method, the priority levels (classes) are solved “nearly separately”, linked only by certain conditional probabilities determined approximately from the solution of other priority levels. Several numerical examples illustrate the accuracy of our approximate solution. The proposed approach significantly reduces the complexity of the problem while featuring generally good accuracy.
Earthquakes can disrupt the healthcare system heavily, leading to long wait times and many untreated patients for years after the event. Emergency services, in particular, must return to preearthquake functionality as soon as possible such that patients, especially critically injured ones, can be treated promptly. However, reconstruction and restoration of emergency services can take years. Due to limited reconstruction resources, decision‐makers cannot reconstruct all hospitals simultaneously. They are typically forced to prioritize the reconstruction order, and this process is often poorly planned. This article models emergency services as an M/M/s queuing system that accounts for prioritized treatment of critical patients and formulates a greedy algorithm to plan for an effective healthcare system reconstruction. The algorithm finds the reconstruction ordering of hospital buildings such that emergency patients have the shortest time to receiving medical care possible. We show our greedy algorithm's good performance for small problem instances, with average deviations from the optimal solution below 16%. Further, we apply our methodology to a case study of Lima, Peru, under a hypothetical M8.0 earthquake. The application demonstrates that compared to typically implemented policies, a policy guided by our formulation results in shorter time to treatment and reduces the number of untreated patients over the course of the reconstruction period by more than a factor of 3 in a worst‐case scenario with 70% hospital capacity disruption. Finally, we demonstrate that our formulation can be integrated into risk analysis through Monte Carlo simulations to inform decision‐makers of reconstruction plans after future earthquakes.
We consider a non-preemptive priority M/M/m queue with two classes of customers and multiple vacations. Service times for all customers are exponentially distributed with the same mean, and vacation times follow an exponential distribution. We obtain the vector probability generating function for the stationary distribution of the number of customers in the queue for each class. This is established by deriving a matrix equation for the vector probability generating function of the stationary distribution of the censored Markov process and then studying the analytical properties of the matrix generating function. We also obtain exact expressions for the first two moments of the number of customers in the queue for each class. Finally, as an application, we investigate a customer’s equilibrium strategy and the optimal priority fee associated with social cost minimization for an unobservable M/M/m queue with two priority classes and multiple vacations.
Emergency department over-crowding has been a growing problem throughout the world. This paper presents a practical approach to estimate the waiting times for multi-class patients and apply the approach to reduce the waiting time for high priority patients. Patient flows with different levels of acuity are formulated based on the priority queue models. It derives explicit expressions of the wait time for the Markov queue and uses the concept of isomorphism to approximate the wait time in the general queue. Numerical results with simulation experiments are reported to display the accuracy of the approach. A case study from an emergency department indicates that the proposed approach can efficiently prioritize patient flows in decreasing waiting times. The queuing models have two features. First, the approximation applies to the general priority queues and reduces to the exact results of the Markov priority queue. Second, the models requires no iterative algorithm.
This paper investigates resource modeling and management for a base station (BS) providing mobile edge computing (MEC) service. In the proposed modeling, BS is recognized as a queueing network consisting of multiple multi-type servers. The uplink transmission (UT) users, downlink transmission (DT) users, and MEC users with different priority levels are jointly considered. It is assumed that their servicerequests arrive dynamically and are also served dynamically. With such a general resource modeling, the interaction among these users can be analyzed based on queueing network theory. The average delay of each service-type with different priority levels is derived. Based on the derived results, two resource management optimization problems are formulated and solved from the perspective of a service provider. The revenue brought by MEC services is first maximized by doing user admission control while provisioning the qualityof- service (QoS) of all admitted users with the given amount of communication and computation resources. Then, the capital expenditure of resource deployment is minimized by satisfying the QoS of all users. It is formulated as an integer programming problem. An algorithm is developed to solve it, which can help service providers to determine the optimal amount of communication and computation resources to be placed in a BS to guarantee QoS for all users at a minimal total capital expenditure. Computer simulations are done to validate all analysis and comparisons are made with BS serving multi-type users of single priority level. Through comparison, an insight is gained that service providers can obtain more revenue or spare less capital expenditure by differentiating the user priority levels.
Cloud computing is known as a new trend for computing resource provision. The process of entering into the cloud is formed as queue, so that each user has to wait until the current user is being served. In this model, the web applications are modeled as queues and the virtual machines are modeled as service centers. M/M/K model is used for multiple priority and multiple server systems with preemptive priorities. To achieve that it distinguish two groups of priority classes that each classes includes multiple items, each having their own arrival and service rate. It derives an approximate method to estimate the steady state probabilities. Based on these probabilities, it can derives approximations for a wide range of relevant performance characteristics, such as the expected postponement time for each item class and the first and second moment of the number of items of a certain type in the system.
We consider a two-stage, multi-server queueing network that serves two types of customers, which we refer to as type a and type b. Type a customers require service at both sequential stages and type b customers only require service at the second stage. The first stage has one node and the second stage has multiple nodes. Type a customers possess a higher non-pre-emptive priority than type b customers. Depending on the model application, two goals are explored: the first goal is to allocate type a customers to the second- stage nodes in a manner that minimizes the average blocking delay; the second goal is to optimize the service speed of each server in the second stage so that the average blocking delay experienced by type a customers is minimized. In this paper, we develop an approximation scheme and an iterative algorithm to find stationary policies, which we then apply to the real-world contexts of Emergency Medical Services planning and airline staffing. Numerical examples show that, compared to some typical heuristic schemes (e.g., proportional allocation based on arrival/service capacity), the suggested allocation policies result in type a customers experiencing shorter delays and allow more of them receive service.
Different virtual machines can share servers, subject to resource constraints. Incoming jobs whose resource requirements cannot be satisfied are queued and receive service according to a preemptive-resume scheduling policy. The problem is to evaluate a cost function, including holding and server costs, with a view to searching for the optimal number of servers. A model with two job types is analyzed exactly and the results are used to develop accurate approximations, which are then extended to more than two classes. Numerical examples and comparisons with simulations are presented.
We consider a two-class two-server retrial queueing system. Customers of each class arrive according to a Poisson process and the service times of each class of customers are assumed to be exponentially distributed with service rates depending on both the customer’s class and the servers. We provide stability and instability conditions for this retrial queueing system.
The algorithm of calculating multi-channel systems with the dynamic priority, which increases linearly with the waiting time. The results of work of the algorithm are given.
The article established a queue model of a preemptive priority queue with impatient customers and multiple vacations. The first class customers have preemptive priority. Both two-class customers can renege because of impatience. First, we obtain State transition equations. Then we derive the probability distribution of the two-class customers ,and respectively calculated the expected number of the two-class customers , the average reneging rate and other queue indices . Finally, we perform analysis by numerical examples.
The diagnosis-stage is always the bottleneck stage for the normal arriving patient flow and the referral patient flow influx in Outpatient Department (OD) in Chinese hospital. And the referral patient has high-priority than the normal arriving patient in general. How to minimize the referral patients mean waiting time and meanwhile keeping the normal patients queue stable? Around the goal a general two types of patient flow with non-preemptive and homogeneous server queueing model is established. An optimal patients routing policies named admissible policy is proposed and the main result formulas are developed. Some analysis around the result is made to verify the validity of the admissible policy. Based on result analysis a new question about server fairness workload is discovered in future research.
Many call centers provide service for customers of different classes with differently qualified groups of agents. Since call arrivals and call handling times are random in inbound call centers, we investigate queueing models for their performance analysis. This paper describes a Markov queueing model with two customer classes and three groups of specialized or flexible agents. The mean call handling time may depend on the respective customer class as well as on the agent group serving this call. The customers are assumed to be impatient. We consider skills-based routing with priority-based rules and derive both steady-state probabilities and performance measures. We present some numerical experiments and show the impact of different mean processing times. The influence of the allocation of the agents into the three groups is analyzed for different levels of patience.
We continue our previous work [Part I, Int. Trans. Oper. Res. 7, No. 6, 653–671 (2000)] to consider an (M,N)-threshold preemptive priority service schedule for a queueing system consisting of two parallel queues and a multiserver. The arrival process for each queue is Poisson, and the service times are exponentially distributed with different means. We derive the generating functions of the stationary joint queue-length distribution, and then obtain the mean queue length and the mean waiting time for each queue.
The stationary queue length distribution for the M/M/1 preemptive priority queue with two classes of customers is studied using the quasi-birth-and-death (QBD) process with infinitely many phases. For the QBD process, we obtain explicit form of the operator-geometric solution such that we can exactly compute its stationary distribution in principle.
The nonpreemptive priority queues are effective for performance evaluations of production/manufacturing systems, inventory control and computer and telecommunication systems. Due to uncontrollable factors, parameters in the nonpreemptive priority queues may be fuzzy. This work constructs the membership functions of the system characteristics of a nonpreemptive priority queueing system with J type jobs, in which the arrival rate and service rate for jobs are all fuzzy numbers. The α-cut approach is used to transform a fuzzy queue into a family of conventional crisp (distinct) queues in this context. By means of the membership functions of the system characteristics, a set of parametric nonlinear programs is developed to describe the family of crisp queues with nonpreemptive priority. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the system characteristics are expressed and governed by the membership functions, the fuzzy nonpreemptive priority queues are represented more accurately and the analytic results are more useful for system designers and practitioners.
We consider a polling system consisting of two-parallel queues and a single server under an (M,N)-threshold nonpreemptive priority service schedule? Two thresholds M and N(0≤M<N) are set up in one of two queuees, say, the second queue. At each epoch of service completion, the server decides which queue is to be served next according to the control level reached by the number of customers in the second queue. For the queueing model, we carry out the performance analysis by using the transform method and propose an algorithm to compute the generating functions of the stationary joint queue length distributions at service completion instants. We also determine the Laplace-Stieltjes transforms of the waiting time distributions for both queues, and obtain their mean waiting times.
The major issue in smart grid (SG) communications networks is the latency management of a vast amount of SG traffic in the access network that connects power substations to a large number of SG monitoring devices. There are three possible deployments for the SG access network considered by power industries: 1) a public access network with a mix of SG and human-to-human (H2H) traffic; 2) a private access network exclusively assigned to SG communications; and 3) a mix of private and public access networks, referred to as hybrid access networks. The SG communications traffic is classified as fixed-scheduling (FS) and event-driven (ED). The FS and ED traffic, generated by SG devices, occur on a periodic basis and as a response to electricity supply conditions, respectively. In this paper, we develop traffic models for public, private, and hybrid SG access networks based on queuing theory. By using these models, we derive an expression for the mean queuing delay for each traffic class in each network. We then propose an optimization problem to find the optimal partitioning of the SG traffic in a hybrid access network. The analytical results obtained from the proposed models agree very well with the simulation results.
This thesis deals with repair capacities and job priorities in repairable spare parts supply systems. Such supply systems can be used to support many different types of technically advanced systems, such as computer systems, medical equipment and military systems. The high price of some components and modules makes repair more profitable than scrap and replace. Upon system failure, it is important that spare parts are available, so that off-line repair is possible, thereby avoiding system down time. A key issue in such systems is to determine adequate spare part stock levels. This is a complex problem, because we generally have to take into account the technical system structure (should we replace modules containing a failed component or should we first disassemble and replace just a component level?) as well as the geographical distribution structure. Hence, we should decide at which level in the product structure spares are needed and in which amount we will store them at which location. Clearly, there is a trade-off between the system availability and spare part investment.
A large range of mathematical models to support such decisions has been developed since the sixties. Most models do not explicitly take into account repair capacities. As a consequence, it is hard to find out how the system performance is influenced by extending or reducing repair capacities. Besides, it is not possible to find out to which extend efficiency gain is possible by proper priority setting. A key idea when starting our research was that we can reduce inventory investment of extremely expensive spare parts by giving them high priority during repair, at the expense of higher stock levels of cheaper items, caused by longer repair throughput times. To find out to which extend this is true, we had to develop more detailed repair shop models than have been used up to now in the literature. When selecting a suitable queueing model for the repair shop, we found out that no suitable results were available in the literature. Therefore, a significant part of the research in this thesis deals with new queueing results that we needed to analyse spare part supply systems with finite repair capacities and repair job priorities. We used these results for three purposes: (1) to examine the impact of finite repair capacities on the spare part inventory optimisation, (2) to make a trade-off between spare part investment and repair capacity investment, and (3) to find rules for repair priority setting. To test and validate our new methods, we conducted extensive numerical results in each step of our research, heavily using discrete event simulation as benchmark.
We consider a multi-class, multi-server queueing system with preemptive priorities. We distinguish two groups of priority classes that consist of multiple customer types, each having their own arrival and service rate. We assume Poisson arrival processes and exponentially distributed service times. We derive an exact method to estimate the steady state probabilities. Because we need iterations to calculate the steady-state probabilities, the only error arises from choosing a finite number of matrix iterations. Based on these probabilities, we can derive approximations for a wide range of relevant performance characteristics, such as the moments of the number of customers of a certain type in the system and the expected postponement time for each customer class. We illustrate our method with some numerical examples. Numerical results show that in most cases we need only a moderate number of matrix iterations (∼20) to obtain an error less than 1% when estimating key performance characteristics.
Customers arriving according to a Markovian arrival process are served at a c server facility. Waiting customers generate into priority while waiting in the system (self-generation of priorities), at a constant rate γ; such a customer is immediately taken for service, if at least one of the servers is free. Else it waits at a waiting space of capacity c exclusively for priority generated customers, provided there is vacancy. A customer in service is not preempted to accommodate a priority generated customer. The service times of ordinary and priority generated customers follow distinct PH-distributions. It is proved that the system is always stable. We provide a numerical procedure to compute the optimal number of servers to be employed to minimize the loss to the system. Several performance measures are evaluated.
This paper deals with multi-server queues with a finite buffer of size N in which units waiting for service generate into priority at a constant rate, independently of other units in the buffer. At the epoch of a unit's priority generation, the unit is immediately taken for service if there is one unit in service that did not generate into priority while waiting; otherwise such a unit leaves the system in search of immediate service elsewhere. The arrival stream of units is a Markovian arrival process (MAP) and service requirements are of phase (PH) type. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a finite quasi-birth-and-death (QBD) process. We give formulas well suited for numerical computation for a variety of performance measures, including the blocking probability, the departure process, and the stationary distributions of the system state at pre-arrival epochs, at post-departure epochs and at epochs at which arriving units are lost. Illustrative numerical examples show the effect of several parameters on certain probabilistic descriptors of the queue for various levels of congestion.
We consider a non-preemptive head-of-the-line multi-server multi-queueing priority model with finite buffer capacity for each priority class. As an arrival process, a generalized Markovian arrival process with marked transitions is used. The service-time distribution is of phase-type and identical for the different priority classes. The model is described by a homogeneous continuous-time Markov chain (CTMC). From the steady-state distribution of the CTMC, which is calculated by matrix-geometric methods, we derive the steady-state distributions immediately after arrival instants of the different priority classes. Applying matrix-analytic methods, we calculate the Laplace-Stieltjes Transform (LST) of the actual waiting times for the different priority classes.
Reliability of mechanical system is closely related to manufacturing quality of the parts, assembly quality of the system, and its working states, including working structure, load, lubricant, vibrancy, temperature, and so on. After the machine manufactured, the design, manufacture and assembly qualities determine the inherent reliability of the products. However, the uncertainty of the working state of the mechanical system at the working stage makes the randomness of the system failure rate of different time segments. This paper builds up the relationship of the system states and its reliability by state-driven reliability model which gives mature consideration of the co-effect of the influencing factors and uncertainty of working state. With this model, this paper gives the analyzing method of system failure rate stochastic characters and simulates the reliability changing curve by Monte Carlo simulation method. Finally, a spindle box of a machine tool was modeled and analyzed as an example. Keywords-NSWC; mechanical system; stochastic state; state- driven reliability modeling; reliability modeling by state-driven reliability model which given mature consideration of the co-effect of influencing factors and uncertainty of working state. The modeling processes was constructed by system kinematics and kinetics modeling, part idleness analyzing, part environment character analyzing, part design and assemble characters analyzing, part failure rate modeling and system failure rate modeling. Through these processes the relationship of system state vector, part state vector, part failure rate and system failure rate can be set up. In many cases, the state of system is constantly changing. Then time discretization method is introduced to deal with this situation. With this model, this paper analyzes stochastic characters of system failure rate and simulates the reliability changing curve and distribution situation by combination of Monte Carlo simulation and time discretization methods. System failure rate stochastic characters are calculated and simulated based on sampling simulation of system character parameters which form the system working state. Then, time dimension is introduced and system reliability changing curve and distribution situation was simulated by Monte Carlo Simulation method. These simulation methods avoid the difficulty of complicated mathematical description and calculation and they are suitable for engineering use. Finally, a spindle box of a machine tool is modeled and analyzed in this paper as an example to demonstrate the reliability modeling and analyzing processes.
We consider a non-preemptive head of the line multi-server priority model. A generalized Markovian arrival process with marked transitions is used as the arrival process. The service-time distribution is of phase-type and identical for the different priority classes. Applying matrix-geometric methods we compute the steady-state distribution of the corresponding single-class system where we do not distinguish between the different priority classes. Furthermore we compute the stationary distribution of the different priority classes immediately after arrival instants. We give an explicit representation for the actual waiting time distribution of the lowest priority customers. Matrix-geometric representations for the Laplace-Stieltjes-Transforms (LST) of the conditional waiting time distributions of different priority classes are derived. Using first passage time analysis and Little's formula, the mean actual waiting times and the mean queue lengths of different priority classes are calculated
We consider a queueing system with multiple servers and two classes of customers operating under a preemptive resume priority rule. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. In a convenient state space representation of the system, we obtain the matrix equation for the two-dimensional, vector-valued generating function of the equilibrium probability distribution. We give a rigorous proof that, by successively eliminating variables from the matrix equations, a nonsingular, block tridiagonal system of equations is obtained for the set of (m+1)(m+2)/2 constants that describe the probabilities of the states of the system when no customers are awaiting service. The mean waiting time of the low priority customers is shown to be given by a simple formula in terms of the (known) waiting time of the high priority customers and the expected number of low priority customers in the queue when no high priority customers are waiting.
We use the matrix-geometric method to study the discrete time MAP/PH/1 priority queue with two types of jobs. Both preemptive and non-preemptive cases are considered. We show that the structure of the R matrix obtained by Miller for the Birth-Death system can be extended to our Quasi-Birth-Death case. For both preemptive and non-preemptive cases the distributions of the number of jobs of each type in the system are obtained and their waiting times are obtained for the non-preemptive. For the preemptive case we obtain the waiting time distribution for the high priority job and the distribution of the lower priority job's wait before it becomes the leading job of its priority class. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 23–50, 1998
The blocking probability of originating calls and forced termination probability of handoff calls are important criteria in performance evaluation of integrated wireless mobile communication networks. In this paper, we model call traffic in a single cell in an integrated voice/data wireless mobile network by a finite buffer queueing system with queueing priority and guard channels. We categorize calls into three types of service classes: originating voice calls, handoff voice calls and data calls. The arrival streams of calls are mutually independent Poisson processes, and channel holding times are exponentially distributed with different means. We describe the behavior of the system by a three-dimensional continuous-time Markov process and present the explicit expression for steady-state distribution of queue lengths using a recursive approach. Furthermore, we calculate the blocking, forced termination probabilities and derive the Laplace–Stieltjes transform of the stationary distribution of actual waiting times and their arbitrary kth moment. Finally, we give some numerical results and discuss the optimization problem for the number of guard channels.
In this paper, we propose an (M, N)-threshold non-preemptive priority service schedule for a queueing system consisting of two-parallel queues and a multi-server. The arrival process for each queue is Poisson, and the service times are exponentially distributed with different means. We derive the generating functions of the stationary joint queue-length distribution, and then obtain the mean queue length and the mean waiting time for each queue.
By using a probabilistic equivalence between the M/G/1 queue with multiple server’s vacations and the M/M/c system, we derive the Laplace- Stieltjes transform of the waiting time W k of a class-k customer in the non-preemptive priority M/M/c queue where all customers have the same mean service time. We also calculate the first two moments of W k .
We study the queue GI/M/s with customers of m different types. An arriving customer is of type i with probability pi
and the types of different customers are independent. A customer of type i requires a service time which is exponentially distributed with parameter bi
. This model is equivalent to the queue GI/Hm/s, where Hm
denotes a mixture of m different exponential distributions. We are primarily interested in the distributions of waiting times and queue lengths. Using a probabilistic argument we reduce the problem to the solution of a system of Wiener-Hopf-type equations. This system is solved by a factorization method. Thus we obtain explicit results for the stationary distributions of waiting times and queue lengths.
In a queuing process, let 1/λ be the mean time between the arrivals of two consecutive units, L be the mean number of units in the system, and W be the mean time spent by a unit in the system. It is shown that, if the three means are finite and the corresponding stochastic processes strictly stationary, and, if the arrival process is metrically transitive with nonzero mean, then L = λW.
A Markov chain is ergodic if it has a proper steady-state distribution. The stability of a queueing system often reduces to the ergodicity of an underlying multidimensional chain. The author gives criteria for the nonergodicity of a multidimensional Markov chain and a generalization to continuous time Markov chains. The author also gives queueing model examples.
There are several commonly occurring situations in which the position of a unit or member of a waiting line is determined by a priority assigned to the unit rather than by its time of arrival in the line. An example is the line formed by messages awaiting transmission over a crowded communication channel in which urgent messages may take precedence over routine ones. With the passage of time a given unit may move forward in the line owing to the servicing of units at the front of the line or may move back owing to the arrival of units holding higher priorities. Though it does not provide a complete description of this process, the average elapsed time between the arrival in the line of a unit of a given priority and its admission to the facility for servicing is useful in evaluating the procedure by which priority assignments are made. Expressions for this quantity are derived for two cases—the single-channel system in which the unit servicing times are arbitrarily distributed (Eq. 3) and the multiple-channel system in which the servicing times are exponentially distributed (Eq. 6). In both cases it is assumed that arrivals occur at random.
Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.
An explicit formula is given for the waiting-time distribution of any number of parallel, negative-exponential servers subject to Poisson demands from any number of priority classes. The demand rates may be different for the priority classes, but the service rate is the same for all classes. The queue discipline is first-come-within-priority class sequence.
A system with N processors and R job classes subject to a preemptive priority scheduling discipline is considered. Exact analysis is presented for the case R = 2, under Markovian assumptions. That analysis suggests a method for obtaining approximate solutions for arbitrary R. The implementation of the methods is discussed and numerical results for some special cases are given.
We study the queue GI/M/s with customers of m different types. An arriving customer is of type i with probability pi and the types of different customers are independent. A customer of type i requires a service time which is exponentially distributed with parameter bi. This model is equivalent to the queue GI/Hm/s, where Hm denotes a mixture of m different exponential distributions. We are primarily interested in the distributions of waiting times and queue lengths. Using a probabilistic argument we reduce the problem to the solution of a system of Wiener-Hopf-type equations. This system is solved by a factorization method. Thus we obtain explicit results for the stationary distributions of waiting times and queue lengths.
The M/G/2 queueing model with service time distribution a mixture of m negative exponential distributions is analysed. The starting point is the functional relation for the Laplace-Stieltjes transform of the stationary joint distribution of the workloads of the two servers. By means of Wiener-Hopf decompositions the solution is constructed and reduced to the solution of m linear equations of which the coefficients depend on the zeros of a polynome. Once this set of equations has been solved the moments of the waiting time distribution can be easily obtained. The Laplace-Stieltjes transform of the stationary waiting time distribution has been derived, it is an intricate expression.
Introduction to Stochastic Processes Priority assignment in waiting line problems
- References Inlar
References (INLAR, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ. COBHAM, A. (1954) Priority assignment in waiting line problems. Operat. Res. 2, 70-76.
Analytic Functions of Several Complex Variables Waiting times in the non-preemptive priority MIMIc queue
- R C Gunning
- H And Rossi
- Nj
- O Kella
- U And Yechiali
GUNNING, R. C. AND RossI, H. (1965) Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs, NJ. KELLA, O. AND YECHIALI, U. (1985) Waiting times in the non-preemptive priority MIMIc queue. Stoch. Models 1, 257-262.