Chapter

# The Remaining Busy Time in a Retrial System with Unreliable Servers

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## Abstract

In this paper, we consider a multiserver retrial queuing system with unreliable servers class-dependent retrial rates and N classes of customers following Poisson input processes. We analyze the distribution of the stationary generalized remaining service time which includes all unavailable periods (setup times) occurring during service of the customer. During service of a class-i customer, the interruptions occur according to the i-dependent Poisson process and the following i-dependent random setup time of the server. We consider two following disciplines caused by the service interruptions: preemptive repeat different and preemptive resume. Using coupling method and regenerative approach, we derive the stationary distribution of the generalized remaining service time in an arbitrary server. For each class i, this distribution is expressed as a convolution of the corresponding original service times and setup times, and in general is available in the terms of the Laplace-Stieltjes transform allowing to calculate the moments of the target distribution. Some numerical examples are included as well.

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Chapter
In this chapter, we extend the analysis developed in Chap. 7 to more complicated retrial systems, in which the retrial rate of each orbit depends on a binary state of the other orbits. This setting is well-motivated and suited for modelling wireless multiple-access systems, see Sect. 8.4.
Conference Paper
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In this work, we investigate the stability conditions of a multiclass retrial system with coupled orbit queues and service interruptions. We consider a single server system accepting N classes of customers according to independent Poisson inputs and with class-dependent, arbitrarily distributed service times. An arriving customer who finds the server unavailable upon arrival, joins the corresponding orbit queue according to its class. We assume that the first (oldest) blocked customer in an orbit queue attempts to connect with the server after an exponentially distributed service time, which depends both on its class, and on the current state (busy or idle) of the other orbit queues. During service times, interruptions occur according to class-dependent Poisson process, following by class-dependent arbitrarily distributed setup times. We consider both preemptive repeat identical, and preemptive-resume interruptions. Potential applications of such a system can be found in the modelling of relay-assisted cooperative wireless networks. We focus on the non-symmetrical orbits and perform simulation experiments for the system with three classes of customers to verify stability conditions for both types of the server interruptions.
Conference Paper
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In this work, we verify by simulation some recent theoretical results describing the dynamics of the the retrial system with coupled orbits. In such a system, retransmission rate of customers blocked in a virtual orbit depends in general on the binary state, busy or idle, of other orbits. We consider a system with N classes of customers, where an arriving customer which meets server busy, joins the corresponding orbit depending on the class of customer. The top (oldest) blocked customer makes an attempt to enter server, with class-dependent exponential time between attempts. At that the retrial rate is defined by the current states (busy or idle) of other orbits. To verify theoretical results, we simulate single-server retrial system with 3 classes of customers following independent Poisson inputs, while service times are class-dependent and have general distributions. In particular, we verify necessary and sufficient stability conditions and focus on the analysis of symmetric model. Numerical experiments confirm theoretical analysis
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We study multi-class retrial queueing systems with Poisson inputs, general service times, and an arbitrary numbers of servers and waiting places. A class-i blocked customer joins orbit i and waits in the orbit for retrial. Orbit i works like a single-server (Formula presented.) queueing system with exponential retrial time regardless of the orbit size. Such retrial systems are referred to as retrial systems with constant retrial rate. Our model is not only motivated by several telecommunication applications, such as wireless multi-access systems, optical networks, and transmission control protocols, but also represents independent theoretical interest. Using a regenerative approach, we provide sufficient stability conditions which have a clear probabilistic interpretation. We show that the provided sufficient conditions are in fact also necessary, in the case of a single-server system without waiting space and in the case of symmetric classes. We also discuss a very interesting case, when one orbit is unstable, whereas the rest of the system is stable.
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This survey paper is devoted to stability analysis of regenerative queueing systems. The method is based on the renewal technique and allows to obtain simple proofs of the sufficient stability conditions which are close to being necessary. The applicability of the method is illustrated by a number of well-known and also less known queueing systems. An extension of the method to the systems under arbitrary initial states is given.
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We consider a GI/G/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has $c$ identical servers and can accommodate the maximal number of $K$ jobs. If a newly arriving job finds the full primary queue, it joins the orbit. The original primary jobs arrive to the system according to a renewal process. The jobs have general i.i.d. service times. A job in front of the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the orbit queue length. Telephone exchange systems, Medium Access Protocols and short TCP transfers are just some applications of the proposed queueing system. For this system we establish minimal sufficient stability conditions. Our model is very general. In addition, to the known particular cases (e.g., M/G/1/1 or M/M/c/c systems), the proposed model covers as particular cases the deterministic service model and the Erlang model with constant retrial rate. The latter particular cases have not been considered in the past. The obtained stability conditions have clear probabilistic interpretation.
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We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.
Article
In this paper, we consider a single-server retrial model with multiple classes of customers. Arrival of customers follow independent Poisson rule. A new customer, facing a busy server upon his arrival, may join the corresponding (class-dependent) orbit queue with a class-dependent probability, or leaves the system forever (balks). The orbit queues follow constant retrial rate discipline, that is, only one (oldest) orbital customer of each orbit queue makes attempts to occupy the server, in a gap of class-dependent exponential times. Within each class, service times are assumed to be independent and identically distributed (iid). We show that this setting generalizes the so-called two-way communication systems. This multiclass system with general service time distributions is analysed using regenerative approach. Necessary and sufficient stability conditions, as well as some explicit expressions for the basic steady-state probabilities, are obtained. A restricted, two-way communication model with exponential service time distributions, is analysed by matrix-analytic method. Moreover, we combine both methods to efficiently derive explicit solutions for the restricted model. An extensive simulation analysis is performed to gain deep insight into the model stability and performance. It is shown that both the simulated and exact results agree on some important measures for which analytical expressions are available, and hence establish the validity of our theoretical treatment. We numerically study the sophisticated dependence structure of the model to uncover the orbits interaction. We give further details and intuitive explanation for the system performance which complements the derived explicit expressions.
Conference Paper
In this work we consider a single-server system accepting $N$ types of retrial customers, which arrive according to independent Poisson streams. In case of blocking, type $i$ customer, $i=1,2,...,N$ is routed to a separate type $i$ orbit queue of infinite capacity. Customers from the orbit queues try to access the server according to the constant retrial policy. We consider coupled orbit queues. More precisely, the orbit queue $i$ retransmits a blocked customer of type $i$ to the main service station after an exponentially distributed time with rate $\mu_{i}$, when at least one other orbit queue is non-empty. Otherwise, if all other orbit queues are empty, the orbit queue $i$ changes its retransmission rate from $\mu_{i}$ to $\mu_{i}^{*}$. Such an operation arises in the modeling of cooperative cognitive wireless networks, in which a node is aware of the status of other nodes, and accordingly, adjusts its retransmission parameters in order to exploit the idle periods of the other nodes. Using the regenerative approach we obtain the necessary conditions of the ergodicity of our system, and show that these conditions have a clear probabilistic interpretation. We also suggest a sufficient stability condition. Simulation experiments show that the obtained conditions delimit the stability domain with remarkable accuracy.
Article
We consider a multiserver retrial queueing system with a renewal input, $K$ classes of customers, and a finite buffer. Service times are class-dependent, however, for each class, are independent, identically distributed (iid). A new class-$i$ customer joins the primary system (servers and buffer), otherwise, if all servers and buffer are full, he joins the class-$i$ (virtual) orbit, and attempts to enter the system after an exponentially distributed time with rate $\gamma_i,\,i=1,\ldots,K$. The retrial discipline is classical because the attempts of different orbital (blocked) customers are independent. We exploit the regenerative structure of the (non-Markovian) queue-size process (total number of customers in the primary system and in orbits) to develop the stability analysis. First we establish the necessary stability conditions, and then show that these conditions are sufficient for stability as well. These conditions coincide with the known stability conditions of a conventional multiclass multiserver system with {\it infinite buffer}. Our analysis covers also the model in which, when a server is free, it makes an outgoing call which occupies the server for a server-dependent random time. Also we extend the stability analysis to the retrial system with a feedback when a served customer returns to the corresponding orbit with a positive probability.
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In this paper we survey work related to queues with interruptions that occur due to many reasons such as server breakdowns, servers taking emergency breaks, and customers having incomplete information or getting distracted. We look at both continuous and discrete time queueing models with interruptions in this survey.
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We study a single server queueing system with general service time distribution and memoryless inter-arrival times. The arrival rate is not constant but varies with the number of customers in the system. A recursive formula for the conditional distribution of the remaining service time given the queue length is derived for an arbitrary epoch. It is also shown that this conditional distribution holds for arrival epochs. The recursion for the corresponding Laplace–Stieltjes transforms and expected values is given in a simple formula.
Article
In many queueing systems, the service process is subject to interruptions resulting from breakdowns, scheduled off-periods or the arrival of customers with preemptive priority. We consider a single server, first-come, first-served queueing system that alternates between periods when service is available (on-periods) and periods when the server is unavailable (off-periods). We consider the case of Poisson arrivals and general service, on- and off-time distributions and derive bounds and approximations for the mean waiting time, probability of delay and steady-state distribution of the number in system. These results are exact for the case of exponentially distributed on-times. Computational results are reported.
Article
We study the customers' Nash equilibrium behavior in a single server observable queue with Poisson arrivals and general service times. Each customer takes a single decision upon arrival: to join or not to join. Furthermore, future regrets are not allowed. The customers are homogenous with respect to their linear waiting cost and the reward associated with service completion. The cost of joining depends on the behavior of the other customers present, which naturally forms a strategic game. We present a recursive algorithm for computing the (possibly mixed) Nash equilibrium strategy. The algorithm's output is queue-dependent joining probabilities. We demonstrate that depending on the service distribution, this equilibrium is not necessarily unique. Also, we show that depending on the service time distribution, either the 'avoid the crowd' phenomenon or the 'follow the crowd' phenomenon may hold.
Article
The distribution of the remaining service time upon reaching some target level in an M/G/1 queue is of theoretical as well as practical interest. In general, this distribution depends on the initial level as well as on the target level, say, B. Two initial levels are of particular interest, namely, level “1” (i.e., upon arrival to an empty system) and level “B−1” (i.e., upon departure at the target level). In this paper, we consider a busy cycle and show that the remaining service time distribution, upon reaching a high level B due to an arrival, converges to a limiting distribution for B→∞. We determine this asymptotic distribution upon the “first hit” (i.e., starting with an arrival to an empty system) and upon “subsequent hits” (i.e., starting with a departure at the target) into a high target level B. The form of the limiting (asymptotic) distribution of the remaining service time depends on whether the system is stable or not. The asymptotic analysis in this paper also enables us to obtain good analytical approximations of interesting quantities associated with rare events, such as overflow probabilities.
Article
The tightness of some queueing stochastic processes is proved and its role in an ergodic analysis is considered. It is proved that the residual service time process in an open Jackson-type network is tight. The same problem is solved for a closed network, where the basic discrete time process is embedded at the service completion epochs. An extention of Kiefer and Wolfowitz's “key” lemma to a nonhomogeneous multiserver queue with an arbitrary initial state is obtained. These results are applied to get the ergodic theorems for the basic regenerative network processes.
On stationary remaining service time in queuieng systems
• E Morozov
• T Morozova