Higher order approximations for tandem queueing networks

In this paper a higher order approximation for single server queues and tandem queueing networks is proposed and studied. Different from the most popular two-moment based approximations in the literature, the higher order approximation uses the higher moments of the interarrival and service distributions in evaluating the performance measures for queueing networks. It is built upon the MacLaurin series analysis, a method that is recently developed to analyze single-node queues, along with the idea of decomposition using higher orders of the moments matched to a distribution. The approximation is computationally flexible in that it can use as many moments of the interarrival and service distributions as desired and produce the corresponding moments for the waiting and interdeparture times. Therefore it can also be used to study several interesting issues that arise in the study of queueing network approximations, such as the effects of higher moments and correlations. Numerical results for single server queues and tandem queueing networks show that this approximation is better than the two-moment based approximations in most cases.     

Limit non-stationary behavior of large closed queueing networks with bottlenecks

In this paper martingales methods are applied for analyzing limit non-stationary behavior of the queue length processes in closed Jackson queueing networks with a single class consisting of a large number of customers, a single infinite server queue, and a fixed number of single server queues with large state independent service rates. It is assumed that one of the single server nodes forms a bottleneck. For the non-bottleneck nodes we show that the queue length distribution at timet converges in generalized sense to the stationary distribution of the M/M/1 queue whose parameters explicitly depend ont. For the bottleneck node a diffusion approximation with reflection is proved in the moderate usage regime while fluid and Gaussian diffusion approximations are established for the heavy usage regime.     

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.     

Queueing Networks of Random Link Topology: Stationary Dynamics of Maximal Throughput Schedules

In this paper, we study the stationary dynamics of a processing system comprised of several parallel queues and a single server of constant rate. The connectivity of the server to each queue is randomly modulated, taking values 1 (connected) or 0 (severed). At any given time, only the currently connected queues may receive service. A key issue is how to schedule the server on the connected queues in order to maximize the system throughput. We investigate two dynamic schedules, which are shown to stabilize the system under the highest possible traffic load, by scheduling the server on the connected queue of maximum backlog (workload or job number). They are analyzed under stationary ergodic traffic flows and connectivity modulation. The results also extend to the more general case of random server rate.We then investigate the dynamics of acyclic (feed-forward) queueing networks with nodes of the previous type. Their links (connectivities) are stochastically modulated, inducing fluctuating network topologies. We focus on the issue of network throughput and show that it is maximized by simple node server schedules. Rate ergodicity of the traffic flows traversing the network is established, allowing the computation of the maximal throughput.Queueing networks of random topology model several practical systems with unreliable service, including wireless communication networks with extraneous interference, flexible manufacturing systems with failing components, production management under random availability of resources etc.Research supported in part by the National Science Foundation.This revised version was published online in June 2005 with corrected coverdate     

On the cycle maximum of birth–death processes and networks of queues

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers in single queues and open Kelly–Whittle networks of queues. For positive recurrent birth–death processes we show that the sequence of sample maxima is stochastically compact. For transient birth–death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact.We show that the Markov chain recording the total number of customers in a Kelly–Whittle network is a birth–death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly–Whittle networks of queues.     

Features of some discrete-time cyclic queueing networks

A class of discrete-time closed cyclic networks is analyzed, where queues at each node have ample waiting room and have independent geometric service times with possibly unequal means. If each node has a single server or if there are sufficiently many parallel servers at each node to accommodate all jobs, equilibrium vectors of product form are obtained. For some other cases, equilibrium vectors of product form need not exist. For the single-server model, a normalization constant is computed and used to determine the queue-length distribution at a node.     

Estimating variance of output from cyclic exponential queueing systems

Production systems, particularly those making use of a pull production control mechanism, are well-modeled as closed queueing networks. Average throughput is clearly one important performance measure for these systems. However, many control decisions require information concerning the variability of the output process as well as throughput. Because of this, the standard deviation of the number of outputs during a specified interval is a practical performance measure for production systems.In this paper, we consider the standard deviation of the number of outputs during a time interval from a closed queueing network consisting ofM single server exponential queues. Because computing this quantity exactly is extremely cumbersome, we introduce a simple approximation that makes use of (1) known results for the variance of the time a marked job takes to complete a round trip and (2) an approximate correction term for the covariance between successive round trips. We show through comparisons with simulation that our method is quite accurate under a variety of conditions.     

Blocking Probabilities in Mobile Communications Networks with Time-Varying Rates and Redialing Subscribers

Call-blocking probabilities are among the key performance measures in mobile communications networks. For their analysis, mobile networks can be modelled as networks of Erlang loss queues with common capacity restrictions dictated by the allocation of frequencies to the cells of the network. However, due to the time-varying load offered to the cells of such networks, blocking probabilities usually cannot be obtained in closed form. The relation between networks of Erlang loss queues and networks of infinite server queues, for which the time-dependent occupancy distribution is multidimensional Poisson, suggests to use that distribution as approximate distribution for the network of Erlang loss queues. This paper extends this so-called Modified Offered Load (MOL) approximation to networks of Erlang loss queues, and also allows subscribers that find their call blocked to redial to continue their call. For GSM networks operating under Fixed Channel Allocation, it is shown that blocking probabilities are increasing in the redial rates so that the MOL approximation that is most accurate for maximal redial rates turns out to be fairly accurate for the resulting upper bound for blocking probabilities. The accuracy is explicitly evaluated in an application of the results towards blocking probabilities in a hot spot travelling along a road through a GSM network.     

Approximate analysis of exponential tandem queues with blocking

In this paper, an approximate method for the analysis of open networks of queues in tandem and with blocking is proposed. The network consists of M single server queuing stations with exogenous Poisson arrival processes and exponentially distributed service times. The analysis is based on the method of decomposition where the total network is broken down into queues which are analyzed as M/C₂/1/N queues assuming Poisson arrival and departure processes to find the steady-state probabilities of the number of customers at each station. The procudure reduces the problem to a number of elementary operations which can be performed efficiently with the aid of a computer. We also compare different definitions of blocking. Numerical results are given to demonstrate the accuracy of the new method.     

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.     

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper, we develop a new method to obtain exact expressions for the Laplace–Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now.     

Transition time asymptotics of queue-based activation protocols in random-access networks

We consider networks where each node represents a server with a queue. An active node deactivates at unit rate. An inactive node activates at a rate that depends on its queue length, provided none of its neighbors is active.For complete bipartite networks, in the limit as the queues become large, we compute the average transition time between the two states where one half of the network is active and the other half is inactive. We show that the law of the transition time divided by its mean exhibits a trichotomy, depending on the activation rate functions.     

Cost sharing of cooperating queues in a Jackson network

We consider networks of queues in which the independent operators of individual queues may cooperate to reduce the amount of waiting. More specifically, we focus on Jackson networks in which the total capacity of the servers can be redistributed over all queues in any desired way. If we associate a cost to waiting that is linear in the queue lengths, it is known from the literature how the operators should share the available service capacity to minimize the long run total cost. This paper deals with the question whether or not (the operators of) the individual queues will indeed cooperate in this way, and if so, how they could share the cost in the new situation such that each operator never pays more than his own cost without cooperation. For the particular case of a tandem network with two or three nodes it is known from previous work that cooperation is indeed beneficial, but for larger tandem networks and for general Jackson networks this question was still open. The main result of this paper gives for any Jackson network an explicit cost allocation that is beneficial for all operators. The approach we use also works for other cost functions, such as the server utilization.     

Interoutput times in processor sharing queues with feedback

While properties of the flows in isolated processor sharing queues are well understood, little is known about the flows in networks with processor sharing nodes. This paper analyzes the internal traffic processes in processor sharing queues with instantaneous Bernoulli feedback. The internal traffic does not inherit the insensitivity to the shape of the service requirement distribution from the external traffic. The interoutput time distribution is studied in the single server and infinite server processor sharing queues. For the systems we study, we show that when service requirement distributions with the same means are convexly ordered, so are interoutput time distributions.This work was partially supported by the National Science Foundation under Grant ECS-8501217 and by the Graduate School of the University of Massachusetts under Faculty Research Grant 1-03205.     

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.     

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.     

Insensitivity in discrete time queues with a moving server

Queues in which customers request service consisting of an integral number of segments and in which the server moves from service station to service station are of considerable interest to practitioners working on digital communications networks. In this paper, we present insensitivity theorems and thereby equilibrium distributions for two discrete time queueing models in which the server may change from one customer to another after completion of each segment of service. In the first model, exactly one segment of service is provided at each time point whether or not an arrival occurs, while in the second model, at most one arrival or service occurs at each time point. In each model, customers of typet request a service time which consists ofl segments in succession with probabilityb _t(l). Examples are given which illustrate the application of the theorems to round robin queues, to queues with a persistent server, and to queues in which server transition probabilities do not depend on the server's previous position. In addition, for models in which the probability that the server moves from one position to another depends only on the distance between the positions, an amalgamation procedure is proposed which gives an insensitive model on a coarse state space even though a queue may not be insensitive on the original state space. A model of Daduna and Schassberger is discussed in this context.This work was supported by the Australian Research Council.     

Two competing discrete-time queues with priority

This paper considers a class of two discrete-time queues with infinite buffers that compete for a single server. Tasks requiring a deterministic amount of service time, arrive randomly to the queues and have to be served by the server. One of the queues has priority over the other in the sense that it always attempts to get the server, while the other queue attempts only randomly according to a rule that depends on how long the task at the head of the queue has been waiting in that position. The class considered is characterized by the fact that if both queues compete and attempt to get the server simultaneously, then they both fail and the server remains idle for a deterministic amount of time. For this class we derive the steady-state joint generating function of the state probabilities. The queueing system considered exhibits interesting behavior, as we demonstrate by an example.     

THE TRANSIENT SOLUTION FOR M/G/1 QUEUEWITH SERVER VACATIONS

1IntroductionDuringrecentseveraldecadesqueueingmodelswitllservervacationshavebeenstudiedin'anumberofpapers(seethecitedreferences).Inthesepaperstwoofthemostremarkableresultsconcernsthestochasticdecompositionpropertiesofqueuelengthandwaitingtimeatarandompointinequilibrium.Butthetransientpropertyoftilequeuelengthforanyqueueingsystemwithservervacationsisnotdiscussedyet.InthispaperwefirstlystlldytheproblemforM/G/1queuewithservervacations.Andthequeueingmodelsconsideredarethefollowing.Model1Conside…