On effects of arrival rate and burstiness in the queueing system: Analysis ofLb/D/1

This paper studies a single-server queueing system with deterministic service time in which arrivals are regulated by the leaky-bucket mechanism. This paper intends to improve quantitative understanding of the effects of arrival rate and burstiness on the average delay of queueing systems. The study is directed toward identifying the worst traffic of arrivals allowed by the leaky-bucket regulation and clarifying the effects of the leaky bucket parameters (which represent the arrival rate and burstiness) on the average queueing delay. The arrival traffic that maximizes the average queueing delay is characterized as the repetition of the following three phases: bulky arrival, greedy arrival for a specified length of interval, and then no arrival till the token bucket is full. The average queueing delay for the worst traffic is expressed as a function the leaky bucket parameters.Research was partially supported by the NSF under grant ECS-8552419. Research was conducted at the Laboratory for Information and Decision Systems of the Massachusetts Institute of Technology and the U.S. Naval Research Laboratory.     

Performance analysis approximation in a queueing system of type M/G/1

In this work, we apply the strong stability method to obtain an estimate for the proximity of the performance measures in the M/G/1 queueing system to the same performance measures in the M/M/1 system under the assumption that the distributions of the service time are close and the arrival flows coincide. In addition to the proof of the stability fact for the perturbed M/M/1 queueing system, we obtain the inequalities of the stability. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results, the STR-STAB algorithm which we execute for a determined queueing system: M/Coxian − 2/1. The accuracy of the approach is evaluated by comparison with simulation results.     

Statistical Analysis of Single-server Loss Queueing Systems

In this article statistical bounds for certain output characteristics of the M/GI/1/n and GI/M/1/n loss queueing systems are derived on the basis of large samples of an input characteristic of these systems, such as service time in the M/GI/1/n queueing system or interarrival time in the GI/M/1/n queueing system. The analysis in this article is based on application of Kolmogorov’s statistics for empirical probability distribution functions.     

Regularly varying tails in a queue with discrete autoregressive arrivals of order p

We consider a discrete time single server queueing system where the service time of a customer is one slot, and the arrival process is governed by a discrete autoregressive process of order p (DAR(p)). For this queueing system, we investigate the tail behavior of the queue size and the waiting time distributions. Specifically, we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β−1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

M x /G/1 Queue with Multiple Vacations

Abstract

In this article, we study a queueing system M ^x /G/1 with multiple vacations. The probability generating function (P.G.F.) of stationary queue length and its expectation expression are deduced by using an embedded Markov chain of the queueing process. The P.G.F. of stationary system busy period and the probability of system in service state and vacation state also are obtained by the same method. At last we deduce the LST and mean of stationary waiting time in the service order FCFS and LCFS, respectively.     

Analysis of a time-limited service priority queueing system with exponential timer and server vacations

We consider a multi-class priority queueing system with a non-preemptive time-limited service controlled by an exponential timer and multiple (or single) vacations. By reducing the service discipline to the Bernoulli schedule, we obtain an expression for the Laplace-Stieltjes transform (LST) of the waiting time distribution via an iteration procedure, and a recursive scheme to calculate the first two moments. It is noted that we have to select embedded Markov points based on the service beginning epochs instead of the service completion epochs adopted for most of M/G/1 queueing analyses. Through the queue-length analysis, we obtain a decomposition form for the LST of the waiting time in each queue having the exhaustive service.      

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

Delay and partial system contents for a discrete-time G-D-c queue

We consider a discrete-time multiserver queueing system with infinite buffer size, constant service times of multiple slots and a first-come-first-served queueing discipline. A relationship between the probability distributions of the partial system contents and the packet delay is established. The relationship is general in the sense that it doesn’t require knowledge of the exact nature of the arrival process. By means of the relationship, results for the distribution of the partial system contents for a wide variety of discrete-time queueing models can be transformed into corresponding results for the delay distribution. As a result, a separate full analysis of the packet delay becomes unnecessary.      

Computational analysis of single-server bulk-arrival queues: GIX/M/1

This paper deals with numerical computations for the bulk-arrival queueing modelGI ^X/M/1. First an algorithm is developed to find the roots inside the unit circle of the characteristic equation for this model. These roots are then used to calculate both the moments and the steady-state distribution of the number of customers in the system at a pre-arrival epoch. These results are used to compute the distribution of the same random variable at post-departure and random epochs. Unifying the method used by Easton [7], we have extended its application to the special cases where the interarrival time distribution is deterministic or uniform, and to cases whereX has a given arbitrary distribution. We also improved on the various root-finding methods used by several previous authors so that high values of the parameters, in particular large batch sizes, can be investigated as well.     

Asymptotic analysis for loss probability of queues with finite <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 type structure

This paper discusses the asymptotic behavior of the loss probability for general queues with finite GI/M/1 type structure such as GI/M/c/K, SM/M/1/K and GI/MSP/1/K queues. We find an explicit expression for the asymptotic behavior of the loss probability as K tends to infinity. With the result, it is shown that the loss probability tends to 0 at a geometric rate. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

Tandem behavior of a telecommunication system with finite buffers and repeated calls

The tandem behavior of a telecommunication system with finite buffers and repeated calls is modeled by the performance of a finite capacityG/M/1 queueing system with general interarrival time distribution, exponentially distributed service time, the first-come-first-served queueing discipline and retrials. In this system a fraction of the units which on arrival at a node of the system find it busy, may retry to be processed, by merging with the incoming arrival units in that node, after a fixed delay time. The performance of this system in steady state is modeled by a queueing network and is approximated by a recursive algorithm based on the isolation method. The approximation outcomes are compared against those from a simulation study. Our numerical results indicate that in steady state the non-renewal superposition arrival process, the non-renewal overflow process, and the non-renewal departure process of the above system can be approximated with compatible renewal processes.     

An M/G/1 queue with cyclic service times

In this paper we consider a single server queue with Poisson arrivals and general service distributions in which the service distributions are changed cyclically according to customer sequence number. This model extends a previous study that used cyclic exponential service times to the treatment of general service distributions. First, the stationary probability generating function and the average number of customers in the system are found. Then, a single vacation queueing system with aN-limited service policy, in which the server goes on vacation after servingN consecutive customers is analyzed as a particular case of our model. Also, to increase the flexibility of using theM/G/1 model with cyclic service times in optimization problems, an approximation approach is introduced in order to obtain the average number of customers in the system. Finally, using this approximation, the optimalN-limited service policy for a single vacation queueing system is obtained.On leave from the Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.     

Comparative analysis for the N policy M/G/1 queueing system with a removable and unreliable server

In this paper we analyze a single removable and unreliable server in the N policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution. The method of maximum entropy is used to develop the approximate steady-state probability distributions of the queue length in the M/G(G)/1 queueing system, where the second and the third symbols denote service time and repair time distributions, respectively. A study of the derived approximate results, compared to the exact results for the M/M(M)/1, M/E₂(E₃)/1, M/H₂(H₃)/1 and M/D(D)/1 queueing systems, suggest that the maximum entropy principle provides a useful method for solving complex queueing systems. Based on the simulation results, we demonstrate that the N policy M/G(G)/1 queueing model is sufficiently robust to the variations of service time and repair time distributions.     

An approximate analysis of a cyclic server queue with limited service and reservations

In this paper, we examine a queueing problem motivated by the pipeline polling protocol in satellite communications. The model is an extension of the cyclic queueing system withM-limited service. In this service mechanism, each queue, after receiving service on cyclej, makes a reservation for its service requirement in cyclej + 1. The main contribution to queueing theory is that we propose an approximation for the queue length and sojourn-time distributions for this discipline. Most approximate studies on cyclic queues, which have been considered before, examine the means only. Our method is an iterative one, which we prove to be convergent by using stochastic dominance arguments. We examine the performance of our algorithm by comparing it to simulations and show that the results are very good.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

A geometric process model for M/M/1 queueing system with a repairable service station

In this paper, we study a geometric process model for M/M/1 queueing system with a repairable service station. By introducing a supplementary variable, some queueing characteristics of the system and reliability indices of the service station are derived. Then a replacement policy N for the service station by which the service station will be replaced following the Nth failure is applied. An optimal replacement policy N¹ for minimizing the long-run average cost per unit time for the service station is then determined.     

A logistics problem: a non-markovian queueing network model and algorithm

In this paper, a stochastic version of the classical deterministic balanced single commodity capacitated transportation network problem is presented. In this model, each arc of the network connects a supply node to a demand node and the flow of units forming along each arc of the network forms a stochastic process (i.e.G/M/1 queueing system with generally distributed interarrival time, a Markovian server, a single server, infinite capacity, and the first come first served queueing discipline). In this model, the total transportation cost is minimized such that the total supply rate is equal to the total demand rate, and the resulting probability of finding excessive congestion along each arc (i.e., the resulting probability of finding congestion inside the queueing system formed along each arc in excess of a fixed number) is equal to a desirable value     

Computing steady-state queueing-time distributions of single-server queues:GI X /M/1

This paper presents a computationally efficient method to find the steady-state distributions of actual queueing times of the first customer, as well as of a randomly selected customer, of an arrival group for the queueing systemGI ^X /M/1, and hence the queueing-time distribution of a customer for the systemGI/E _X /1. The distribution of virtual queueing time is also obtained. Approximate analysis based on one or more roots is also discussed. Though the exact detailed as well as approximate computations for a variety of interarrival-time distributions such as generalized Erlang, mixed generalized Erlang, hyperexponential, generalized hyperexponential, and deterministic have been carried out, only representative results in the form of tables have been appended. The results obtained should prove useful to queueing theorists, practitioners, and others.     

Dynamic admission and service rate control of a queue

This paper investigates a queueing system in which the controller can perform admission and service rate control. In particular, we examine a single-server queueing system with Poisson arrivals and exponentially distributed services with adjustable rates. At each decision epoch the controller may adjust the service rate. Also, the controller can reject incoming customers as they arrive. The objective is to minimize long-run average costs which include: a holding cost, which is a non-decreasing function of the number of jobs in the system; a service rate cost c(x), representing the cost per unit time for servicing jobs at rate x; and a rejection cost κ for rejecting a single job. From basic principles, we derive a simple, efficient algorithm for computing the optimal policy. Our algorithm also provides an easily computable bound on the optimality gap at every step. Finally, we demonstrate that, in the class of stationary policies, deterministic stationary policies are optimal for this problem.