A single server queue with service interruptions

In this paper we characterize the queue-length distribution as well as the waiting time distribution of a single-server queue which is subject to service interruptions. Such queues arise naturally in computer and communication problems in which customers belong to different classes and share a common server under some complicated service discipline. In such queues, the viewpoint of a given class of customers is that the server is not available for providing service some of the time, because it is busy serving customers from a different class. A natural special case of these queues is the class of preemptive priority queues. In this paper, we consider arrivals according the Markovian Arrival Process (MAP) and the server is not available for service at certain times. The service times are assumed to have a general distribution. We provide numerical examples to show that our methods are computationally feasible. This revised version was published online in June 2006 with corrections to the Cover Date.     

A finite capacity multi-queueing system with priorities and with repeated server vacations

In this paper, we study an M/G/1 multi-queueing system consisting ofM finite capacity queues, at which customers arrive according to independent Poisson processes. The customers require service times according to a queue-dependent general distribution. Each queue has a different priority. The queues are attended by a single server according to their priority and are served in a non-preemptive way. If there are no customers present, the server takes repeated vacations. The length of each vacation is a random variable with a general distribution function. We derive steady state formulas for the queue length distribution and the Laplace transform of the queueing time distribution for each queue.     

Conditional and unconditional distributions forM/G/1 type queues with server vacations

M/G/1 queues with server vacations have been studied extensively over the last two decades. Recent surveys by Boxma [3], Doshi [5] and Teghem [14] provide extensive summary of literature on this subject. More recently, Shanthikumar [11] has generalized some of the results toM/G/1 type queues in which the arrival pattern during the vacations may be different from that during the time the server is actually working. In particular, the queue length at the departure epoch is shown to decompose into two independent random variables, one of which is the queue length at the departure epoch (arrival epoch, steady state) in the correspondingM/G/1 queue without vacations. Such generalizations are important in the analysis of situations involving reneging, balking and finite buffer cyclic server queues. In this paper we consider models similar to the one in Shanthikumar [11] but use the work in the system as the starting point of our investigation. We analyze the busy and idle periods separately and get conditional distributions of work in the system, queue length and, in some cases, waiting time. We then remove the conditioning to get the steady state distributions. Besides deriving the new steady state results and conditional waiting time and queue length distributions, we demonstrate that the results of Boxma and Groenendijk [2] follow as special cases. We also provide an alternative approach to deriving Shanthikumar's [11] results for queue length at departure epochs.     

A two-queue model with Bernoulli service schedule and switching times

In this paper, we present a detailed analysis of a cyclic-service queueing system consisting of two parallel queues, and a single server. The server serves the two queues with a Bernoulli service schedule described as follows. At the beginning of each visit to a queue, the server always serves a customer. At each epoch of service completion in the ith queue at which the queue is not empty, the server makes a random decision: with probability p_i, it serves the next customer; with probability 1-p_i, it switches to the other queue. The server takes switching times in its transition from one queue to the other. We derive the generating functions of the joint stationary queue-length distribution at service completion instants, by using the approach of the boundary value problem for complex variables. We also determine the Laplace-Stieltjes transforms of waiting time distributions for both queues, and obtain their mean waiting times. This revised version was published online in June 2006 with corrections to the Cover Date.     

Waiting time analysis of the multiple priority dual queue with a preemptive priority service discipline

The dual queue consists of two queues, called the primary queue and the secondary queue. There is a single server in the primary queue but the secondary queue has no service facility and only serves as a holding queue for the overloaded primary queue. The dual queue has the additional feature of a priority scheme to help reduce congestion. Two classes of customers, class 1 and 2, arrive to the dual queue as two independent Poisson processes and the single server in the primary queue dispenses an exponentially distributed service time at the rate which is dependent on the customer’s class. The service discipline is preemptive priority with priority given to class 1 over class 2 customers. In this paper, we use matrix-analytic method to construct the infinitesimal generator of the system and also to provide a detailed analysis of the expected waiting time of each class of customers in both queues.     

An explicit solution to a tandem queueing model

We consider two queues in tandem, each with an exponential server, and with deterministic arrivals to the first queue. We obtain an explicit solution for the steady state distribution of the process (N₁(t), N₂(t), Y(t)), where N_j(t) is the queue length in the jth queue and Y(t) measures the time elapsed since the last arrival. Then we obtain the marginal distributions of (N₁(t), N₂(t)) and of N₂(t). We also evaluate the solution in various limiting cases, such as heavy traffic. This revised version was published online in June 2006 with corrections to the Cover Date.     

Factorization and Stochastic Decomposition Properties in Bulk Queues with Generalized Vacations

This paper considers a class of stationary batch-arrival, bulk-service queues with generalized vacations. The system consists of a single server and a waiting room of infinite capacity. Arrivals of customers follow a batch Markovian arrival process. The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in groups of fixed size B. For this class of queues, we show that the vector probability generating function of the stationary queue length distribution is factored into two terms, one of which is the vector probability generating function of the conditional queue length distribution given that the server is on vacation. The special case of batch Poisson arrivals is carefully examined, and a new stochastic decomposition formula is derived for the stationary queue length distribution.AMS subject classification: 60K25, 90B22, 60K37     

Quasi-reversible multiclass queues with order independent departure rates

This paper introduces a new class of queues which are quasi-reversible and therefore preserve product form distribution when connected in multinode networks. The essential feature leading to the quasi-reversibility of these queues is the fact that the total departure rate in any queue state is independent of the order of the customers in the queue. We call such queues order independent (OI) queues. The OI class includes a significant part of Kelly's class of symmetric queues, although it does not cover the whole class. A distinguishing feature of the OI class is that, among others, it includes the MSCCC and MSHCC queues but not the LCFS queue. This demonstrates a certain generality of the class of OI queues and shows that the quasi-reversibility of the OI queues derives from causes other than symmetry principles. Finally, we examine OI queues where arrivals to the queue are lost when the number of customers in the queue equals an upper bound. We obtain the stationary distribution for the OI loss queue by normalizing the stationary probabilities of the corresponding OI queue without losses. A teletraffic application for the OI loss queue is presented.     

Stationary tail probabilities in exponential server tandems with renewal arrivals

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths of the arrival and virtual service processes in order to get a large queue size at the second queue. We then use large deviations estimates of the probabilities of these paths, and solve a constrained convex optimization problem to find the most likely path leading to a large queue size. We find that the stationary queue length distribution at the second queue has an exponentially decaying tail, and obtain the exact rate of decay.Research supported in part by NSF grant NCR 88-57731 and the AT & T Foundation.     

A two-queue,one-server model with priority for the longer queue

The queueing model studied consists of one server and two queues. Each queue has its own Poisson arrival stream and service time distribution. After a service completion, the server proceeds with a customer from the longer queue, if the queues are unequal; if the queues are equal, the server chooses with some probability a customer from one of the queues. The model is of practical interest in performance analysis, but also of theoretical interest because the functional equation to be solved has not yet been studied in the queueing literature. A basic analysis of this functional equation is presented. Some numerical results are given to assess the influence of the present service discipline. Some new properties of L.S. transforms of service time distributions are discussed in the appendix.Dr. T. Katayama has formulated the present problem and brought it to the author's attention during his visit in October/November 1984 to the NTT-Electr. Comm. Lab.'s Musashino, Tokyo 180.     

Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs

Retrial queues have been widely used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, etc. It is of basic importance to study reliability of retrial queues with server breakdowns and repairs because of limited ability of repairs and heavy influence of the breakdowns on the performance measure of the system. However, so far the repairable retrial queues are analyzed only by queueing theory. In this paper we give a detailed analysis for reliability of retrial queues. By using the supplementary variables method, we obtain the explicit expressions of some main reliability indexes such as the availability, failure frequency and reliability function of the server. In addition, some special queues, for instance, the repairable M/G/1 queue and repairable retrial queue can be derived from our results. These results may be generalized to the repairable multi-server retrial models.     

A Markovian single server with upstream job and downstream demand arrival stream

In this paper we consider a Markovian single server system which processes items arriving from an upstream region (as usual in queueing systems) and is controlled by a demand arrival stream for finished items from a downstream area. A finite storage is available at the server to store finished items not immediately needed in the downstream area. The system considered corresponds to an assembly-like queue with two input streams. The system is stable in a strict sense only if all queues are finite, i.e., both random processes are synchronized via blocking. This notion leads to a complementary system with a very similar state space which is a pair of Markovian single servers with synchronous arrivals. In the mathematical analysis the main focus is on the state probabilities and expectation of minimum and maximum of the two input queues. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.     

A survey on retrial queues

Queueing systems in which arriving customers who find all servers and waiting positions (if any) occupied may retry for service after a period of time are called retrial queues or queues with repeated orders. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication networks, computer networks and computer systems. In this paper, we discuss some important retrial queueing models and present their major analytic results and the techniques used. Our concentration is mainly on single-server queueing models. Multi-server queueing models are briefly discussed, and interested readers are referred to the original papers for details. We also discuss the stochastic decomposition property which commonly holds in retrial queues and the relationship between the retrial queue and the queue with server vacations.     

Relating polling models with zero and nonzero switchover times

We consider a system ofN queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues), and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the next). We consider two versions of this classical polling model: In the first, which we refer to as the zero-switchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzero-switchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann [Queueing Systems 11 (1992) 109—120] and Cooper, Niu, and Srinivasan [to appear in Oper. Res.].Research supported in part by the National Science Foundation under grant DDM-9001751.     

A polling model with an autonomous server

This paper considers polling systems with an autonomous server that remains at a queue for an exponential amount of time before moving to a next queue incurring a generally distributed switch-over time. The server remains at a queue until the exponential visit time expires, also when the queue becomes empty. If the queue is not empty when the visit time expires, service is preempted upon server departure, and repeated when the server returns to the queue. The paper first presents a necessary and sufficient condition for stability, and subsequently analyzes the joint queue-length distribution via an embedded Markov chain approach. As the autonomous exponential visit times may seem to result in a system that closely resembles a system of independent queues, we explicitly investigate the approximation of our system via a system of independent vacation queues. This approximation is accurate for short visit times only.      

Markovian queues with Poisson control

We investigate Markovian queues that are examined by a controller at random times determined by a Poisson process. Upon examination, the controller sets the service speed to be equal to the minimum of the current number of customers in the queue and a certain maximum service speed; this service speed prevails until the next examination time. We study the resulting two-dimensional Markov process of queue length and server speed, in particular two regimes with time scale separation, specifically for infinitely frequent and infinitely long examination times. In the intermediate regime the analysis proves to be extremely challenging. To gain further insight into the model dynamics we then analyse two variants of the model in which the controller is just an observer and does not change the speed of the server.     

A single server queue in a hard-real-time environment

We consider a single server first in first out queue in which each arriving task has to be completed within a certain period of time (its deadline). More precisely, each arriving task has its own deadline - a non-negative real number - and as soon as the response time of one task exceeds its deadline, the whole system in considered to have failed. (In that sense the deadline is hard.) The main practical motivation for analyzing such queues comes from the need to evaluate mathematically the reliability of computer systems working with real time constraints (space or aircraft systems for instance). We shall therefore be mainly concerned with the analytical characterization of the transient behavior of such a queue in order to determine the probability of meeting all hard deadlines during a finite period of time (the ‘mission time’). The probabilistic methods for analyzing such systems are suggested by earlier work on impatience in telecommunication systems [1,2].     

Vacation models in discrete time

A class of single server vacation queues which have single arrivals and non-batch service is considered in discrete time. It is shown that provided the interarrival, service, vacation, and server operational times can be cast with Markov-based representation then this class of vacation model can be studied as a matrix–geometric or a matrix-product problem – both in the matrix–analytic family – thereby allowing us to use well established results from Neuts (1981). Most importantly it is shown that using discrete time approach to study some vacation models is more appropriate and makes the models much more algorithmically tractable. An example is a vacation model in which the server visits the queue for a limited duration. The paper focuses mainly on single arrival and single unit service systems which result in quasi-birth-and-death processes. The results presented in this paper are applicable to all this class of vacation queues provided the interarrival, service, vacation, and operational times can be represented by a finite state Markov chain.An erratum to this article can be found at     

A heavy-traffic analysis of a closed queueing system with a GI/∞ service center

This paper studies the heavy-traffic behavior of a closed system consisting of two service stations. The first station is an infinite server and the second is a single server whose service rate depends on the size of the queue at the station. We consider the regime when both the number of customers, n, and the service rate at the single-server station go to infinity while the service rate at the infinite-server station is held fixed. We show that, as n→∞, the process of the number of customers at the infinite-server station normalized by n converges in probability to a deterministic function satisfying a Volterra integral equation. The deviations of the normalized queue from its deterministic limit multiplied by √n converge in distribution to the solution of a stochastic Volterra equation. The proof uses a new approach to studying infinite-server queues in heavy traffic whose main novelty is to express the number of customers at the infinite server as a time-space integral with respect to a time-changed sequential empirical process. This gives a new insight into the structure of the limit processes and makes the end results easy to interpret. Also the approach allows us to give a version of the classical heavy-traffic limit theorem for the G/GI/∞ queue which, in particular, reconciles the limits obtained earlier by Iglehart and Borovkov. This revised version was published online in June 2006 with corrections to the Cover Date.