Using the M/G/1 queue under processor sharing for exact simulation of queues

In Sigman (J. Appl. Probab. 48A:209–216, 2011b), a first exact simulation algorithm was presented for the stationary distribution of customer delay for FIFO M/G/c queues in which ρ=λ/μ<1 (super stable case). The key idea involves dominated coupling from the past while using the M/G/1 queue under the processor sharing (PS) discipline as a sample-path upper bound, taking advantage of its time-reversibility properties so as to be able to simulate it backwards in time. Here, we expand upon this method and give several examples of other queueing models for which this method can be used to exactly simulate from their stationary distributions. Examples include sojourn times for single-server queues under various service disciplines, tandem queues, and multi-class networks with general routing.     

An M/G/1 queue under hysteretic vacation policy with an early startup and un-reliable server

This paper considers the bi-level control of an M/G/1 queueing system, in which an un-reliable server operates N policy with a single vacation and an early startup. The server takes a vacation of random length when he finishes serving all customers in the system (i.e., the system is empty). Upon completion of the vacation, the server inspects the number of customers waiting in the queue. If the number of customers is greater than or equal to a predetermined threshold m, the server immediately performs a startup time; otherwise, he remains dormant in the system and waits until m or more customers accumulate in the queue. After the startup, if there are N or more customers waiting for service, the server immediately begins serving the waiting customers. Otherwise the server is stand-by in the system and waits until the accumulated number of customers reaches or exceeds N. Further, it is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. We obtain the probability generating function in the system through the decomposition property and then derive the system characteristics     

On the convexity of the two-threshold policy for an M/G/1 queue with vacations

In this note, we consider a single server queueing system with server vacations of two types and a two-threshold policy. Under a cost and revenue structure the long-run average cost function is proven to be convex in the lower threshold for a fixed difference between the two thresholds.     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

带有Bernoulli控制策略的M/M/1多重休假排队模型

研究具有Bernoulli控制策略的M/M/1多重休假排队模型: 当系统为空时, 服务台依一定的概率或进入闲期, 或进入普通休假状态, 或进入工作休假状态. 对该模型, 应用拟生灭(QBD)过程和矩阵几何解的方法, 得到了过程平稳队长的具体形式, 在此基础上, 还得到了平稳队长和平稳逗留时间的随机分解结果以及附加队长分布和附加延迟的LST的具体形式. 结果表明, 经典的M/M/1排队, M/M/1多重休假排队, M/M/1多重工作休假排队都是该模型的特殊情形.     

M/G/1 queue with controllable vacations and optimization of vacation policy

We introduce the control parameterN in a common queue M/G/1 with vacations; the end of a global vacation period is controlled by the parameterN. This extension for a queue with vacations is of significance in certain practical cases. In this paper, we find various transient and steady-state results for the queue size, the delay times and the waiting times for the M/G/1 queue with controllable vacations. Finally, we also discuss optimal selection of the control parameter.     

Analysis of the M/G/1 processor-sharing queue with bulk arrivals

We analyze the single server processor-sharing queue for the case of bulk arrivals. We obtain an expression for the expected response time of a job as a function of its size, when the service times of jobs have a generalized hyperexponential distribution and more generally for distributions with rational Laplace transforms. Our analysis significantly extends the class of distributions for which processor-sharing queues with bulk arrivals were previously analyzed.     

A polynomial-time algorithm for computing an optimal admission policy in a GI/M/1/N queue

A polynomial-time algorithm is proposed for computing an optimal admission policy for GI/M/1/N queueing systems. The approach is based upon mathematical programming in which a pointwise minimal value functions is to be found subject to a finite number of DP type constraints. The program is transformed by Fourier-Motzkin elimination into equivalent reduced systems to which a simple, forward-substitution type algorithm can be applied.     

Tail asymptotics for the queue length in an M/G/1 retrial queue

In this paper, we study the tail behavior of the stationary queue length of an M/G/1 retrial queue. We show that the subexponential tail of the stationary queue length of an M/G/1 retrial queue is determined by that of the corresponding M/G/1 queue, and hence the stationary queue length in an M/G/1 retrial queue is subexponential if the stationary queue length in the corresponding M/G/1 queue is subexponential. Our results for subexponential tails also apply to regularly varying tails, and we provide the regularly varying tail asymptotics for the stationary queue length of the M/G/1 retrial queue. AMS subject classifications: 60J25, 60K25     

Optimal control of a removable server in an M/G/1 queue with finite capacity

We consider an M/G/1 system with finite capacity L in which the removable server applies a (ν, N) policy; a classical cost structure is imposed and the total expected cost per unit time in the steady state is considered. For the M/M/1 situation, Hersh and Brosh [3] analysed the policies with 0⩾ν<N⩽L and established that the best of them is characterized either by ν = 0 or by N = L. By a different and quite easy way and for a general service time distribution, we prove that an optimal policy has the form (ν = 0, 0 ⩽ N ⩽ L + 1), where the (0, 0) and (0, L + 1) policies consist in never closing or never opening the station respectively. Moreover, we describe a precise technique to analyse the policy space and to determine easily the optimal policy.     

Analysis of an M/G/1 queue with vacations and multiple phases of operation

This paper deals with an M / G / 1 queue with vacations and multiple phases of operation. If there are no customers in the system at the instant of a service completion, a vacation commences, that is, the system moves to vacation phase 0. If none is found waiting at the end of a vacation, the server goes for another vacation. Otherwise, the system jumps from phase 0 to some operative phase i with probability \(q_i\), \(i = 1,2, \ldots ,n.\) In operative phase i, \(i = 1,2, \ldots ,n\), the server serves customers according to the discipline of FCFS (First-come, first-served). Using the method of supplementary variables, we obtain the stationary system size distribution at arbitrary epoch. The stationary sojourn time distribution of an arbitrary customer is also derived. In addition, the stochastic decomposition property is investigated. Finally, we present some numerical results.     

AnM/M/1 retrial queue with control policy and general retrial times

We consider anM/M/1 retrial queueing system in which the retrial time has a general distribution and only the customer at the head of the queue is allowed to retry for service. We find a necessary and sufficient condition for ergodicity and, when this is satisfied, the generating function of the distribution of the number of customers in the queue and the Laplace transform of the waiting time distribution under steady-state conditions. The results agree with known results for special cases.Supported by KOSEF 90-08-00-02.     

A discrete-time Geo[X]/G/1 retrial queue with control of admission

This paper analyses a discrete-time Geo/G/1 retrial queue with batch arrivals in which individual arriving customers have a control of admission. We study the underlying Markov chain at the epochs immediately after the slot boundaries making emphasis on the computation of its steady-state distribution. To this end we employ numerical inversion and maximum entropy techniques. We also establish a stochastic decomposition property and prove that the continuous-time M/G/1 retrial queue with batch arrivals and control of admission can be approximated by our discrete-time system. The outcomes agree with known results for special cases.     

M/G/1 queue with single working vacation

In this paper, an M/G/1 queue with single working vacation is analyzed. Using the method of supplementary variable and the matrix-analytic method, we obtain the queue length distribution and service status at the arbitrary epoch under steady state conditions. Further, we derive expected busy period and expected busy cycle. Finally, server special cases are presented.     

M/G/1 queue with event-dependent arrival rates

Motivated by experiments on customers’ behavior in service systems, we consider a queueing model with event-dependent arrival rates. Customers’ arrival rates depend on the last event, which may either be a service departure or an arrival. We derive explicitly the performance measures and analyze the impact of the event-dependency. In particular, we show that this queueing model, in which a service completion generates a higher arrival rate than an arrival, performs better than a system in which customers are insensitive to the last event. Moreover, contrary to the M/G/1 queue, we show that the coefficient of variation of the service does not necessarily deteriorate the system performance. Next, we show that this queueing model may be the result of customers’ strategic behavior when only the last event is known. Finally, we investigate the historical admission control problem. We show that, under certain conditions, a deterministic policy with two thresholds may be optimal. This new policy is easy to implement and provides an improvement compared to the classical one-threshold policy.     

The G/M/1 queue revisited

The G/M/1 queue is one of the classical models of queueing theory. The goal of this paper is two-fold: (a) To introduce new derivations of some well-known results, and (b) to present some new results for the G/M/1 queue and its variants. In particular, we pay attention to the G/M/1 queue with a set-up time at the start of each busy period, and the G/M/1 queue with exceptional first service. Dedicated to Arie Hordijk on his 65th birthday, in friendship and admiration.     

Equilibrium properties of the M/G/1 queue

Summary Various aspects of the equilibrium M/G/1 queue at large values are studied subject to a condition on the service time distribution closely related to the tail to decrease exponentially fast. A simple case considered is the supplementary variables (age and residual life of the current service period), the distribution of which conditioned upon queue length n is shown to have a limit as n. Similar results hold when conditioning upon large virtual waiting times. More generally, a number of results are given which describe the input and output streams prior to large values e.g. in the sense of weak convergence of the associated point processes and incremental processes. Typically, the behaviour is shown to be that of a different transient M/G/1 queueing model with a certain stochastically larger service time distribution and a larger arrival intensity. The basis of the asymptotic results is a geometrical approximation for the tail of the equilibrium queue length distribution, pointed out here for the GI/G/1 queue as well.     

Some exact expressions for the bulk-arrival queue mx/M/1

For a M/M/1 queueing system with group arrivals of random size the transition probabilities of the queue size process and the distribution of the maximal queue size during a time interval [0,t) are calculated. Simple formulae for the corresponding Laplace transforms are given.     

The M/M/1+M queue with a utility-maximizing server

We consider an M/M/1+M queue with a human server, who is influenced by incentives. Specifically, the server chooses his service rate by maximizing his utility function. Our objective is to guarantee the existence of a unique maximum. The complication is that most sensible utility functions depend on the server utilization, a non-simple expression. We derive a property of the utilization that guarantees quasiconcavity of any utility function that multiplies the server’s concave (including linear) “value” of his service rate by the server utilization.     

On the M/G/1 foreground-background processor-sharing queue

We consider the M/G/1 queue under the foreground-background processor-sharing discipline. Using a result on the stationary distribution of the total number of customers we give a direct derivation of the distribution of the random counting measure representing the steady state of the queue in all detail.This work was done during a sabbatical at INRIA, France.