Optimal Control of an M/G/1/K Queueing System with Combined <Emphasis Type="Italic">F</Emphasis> Policy and Startup Time

We investigate the optimal management problem of an M/G/1/K queueing system with combined F policy and an exponential startup time. The F policy queueing problem investigates the most common issue of controlling the arrival to a queueing system. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in the system. The method is illustrated analytically for exponential service time distribution. A cost model is established to determine the optimal management F policy at minimum cost. We use an efficient Maple computer program to calculate the optimal value of F and some system performance measures. Sensitivity analysis is also investigated.     

A NEW APPROACH FOR THE STUDY OF THE M X /G/1 QUEUE USING RENEWAL ARGUMENTS

The M ^X /G/1 queueing system as well as several of its variants have long ago been studied by considering the embedded discrete-time Markov chain at service completion epochs. Alternatively other approaches have been proposed such as the theory of regenerative processes, the supplementary variable method, properties of the busy period, etc. In this note we study the M ^X /G/1 queue via a simple new method that uses renewal arguments. This approach seems quite powerful and may become fruitful in the investigation of other queueing systems as well.     

Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue

In this paper, we study the stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. We introduce an embedded Markov chain of matrix M/G/1 type with a tree structure and identify conditions for the Markov chain to be ergodic. First, we present three conventional methods for the stability problem of the queueing system of interest. These methods are either computationally demanding or do not provide accurate information for system stability. Then we introduce a novel approach that develops two linear programs whose solutions provide sufficient conditions for stability or instability of the queueing system. The new approach is numerically efficient. The advantages and disadvantages of the methods introduced in this paper are analyzed both theoretically and numerically.     

Strong Stability of the Batch Arrival Queueing Systems

In this work, we use the strong stability method to study the batch arrival queue after a perturbation of the batch size distribution. We show that, under some hypotheses, the characteristics of the batch arrival queueing system M ^X /M/1 may be approximated by the correspondent characteristics of the system M ^Geo /M/1.

After clarifying the conditions of approximation, we obtain stability inequalities with an exact computation of constants.     

An MX/G/1 queueing system with a setup period and a vacation period

This paper deals with an M^X/G/1 queueing system with a vacation period which comprises an idle period and a random setup period. The server is turned off each time when the system becomes empty. At this point of time the idle period starts. As soon as a customer or a batch of customers arrive, the setup of the service facility begins which is needed before starting each busy period. In this paper we study the steady state behaviour of the queue size distributions at stationary (random) point of time and at departure point of time. One of our findings is that the departure point queue size distribution is the convolution of the distributions of three independent random variables. Also, we drive analytically explicit expressions for the system state probabilities and some performance measures of this queueing system. Finally, we derive the probability generating function of the additional queue size distribution due to the vacation period as the limiting behaviour of the M^X/M/1 type queueing system. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of the M x /G/1 Queue with a Random Setup Time

Abstract

We consider an M ^x /G/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An M ^x /G/1 queueing system with setup period and a vacation period. Queueing Sys. 2000, 36, 23–38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury4 Choudhury, G. 2000. An M^x/G/1 queueing system with setup period and a vacation period. Queueing Syst., 36: 23–38. [CROSSREF][Crossref], [Web of Science ®] , [Google Scholar]. Finally, we present a transform free method to obtain the mean waiting time of this model.     

The M/G/1 Queue with Finite Workload Capacity

We consider the M/G/1 queueing system in which customers whose admission to the system would increase the workload beyond a prespecified finite capacity limit are not accepted. Various results on the distribution of the workload are derived; in particular, we give explicit formulas for its stationary distribution for M/M/1 and in the general case, under the preemptive LIFO discipline, for the joint stationary distribution of the number of customers in the system and their residual service times. Furthermore, the Laplace transform of the length of a busy period is determined. Finally, for M/D/1 the busy period distribution is derived in closed form.     

Asymptotic Analysis of the GI/M/ 1 /n Loss System as n Increases to Infinity

This paper provides the asymptotic analysis of the loss probability in the GI/M/1/n queueing system as n increases to infinity. The approach of this paper is alternative to that of the recent papers of Choi and Kim (2000) and Choi et al. (2000) and based on application of modern Tauberian theorems with remainder. This enables us to simplify the proofs of the results on asymptotic behavior of the loss probability of the abovementioned paper of Choi and Kim (2000) as well as to obtain some new results.     

Truncated Erlangian queueing system with fuzzy arrival rate,balking and reneging

In this paper, we derive the performance measures of the truncated Erlangian service queueing system with state-dependent rate, balking and reneging with fuzzy arrival rate _n\tilde \lambda _n so for the (M/E _r /1/N(α, β)), we have obtained P _n,s the steady state probabilities in the system with the unit in the service being at stage s (s=1, 2, …, r); k is the boundary state for the number of customer more than the one that the rate of service increases. We treat this queueing system for general values of r, k and N and derive the Fuzzy effective measures for the operation of the system at any time.     

Delay analysis of extended rtPS for VoIP service in IEEE 802.16e by matrix analytic method

Extended real time polling service (ErtPS) is added to IEEE 802.16e-2005 standards in order for VoIP service to use uplink resources efficiently by considering on/off characteristic of voice source. Recently average queueing delay of ErtPS algorithm for VoIP service was investigated, and it was shown that ErtPS allows to admit more users than UGS algorithm. But we need the probability distribution of queueing delay rather than average queueing delay in order to provide a necessary information for QoS. In this paper we obtain the probability distribution of queueing delay of ErtPS for VoIP service by using the matrix analytic method for the GI/M/1 type and the M/G/1 type matrices in cases of the service time being exponential and deterministic respectively. By applying the results on deterministic service time we find the maximum allowable number of VoIP users with the required constraint on queueing delay. 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).     

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.     

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.     

Buffer allocation for a class of nonlinear stochastic knapsack problems

In this paper, we examine a class of nonlinear, stochastic knapsack problems which occur in manufacturing, facility or other network design applications.Series, merge-and-split topologies of series-parallelM/M/1/K andM/M/C/K queueing networks with an overall buffer constraint bound are examined. Bounds on the objective function are proposed and a sensitivity analysis is utilized to quantify the effects of buffer variations on network performance measures.     

Transient analysis of a single server queue with catastrophes, failures and repairs

A transient solution is obtained analytically using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and non-zero repair time. The steady state probability of the system size is present. Some key performance measures, namely, throughput, loss probability and response time for the system under consideration are investigated. Further, reliability and availability of the system are analysed. Finally, numerical illustrations are used to discuss the system performance measures.      

TheM/G/1 retrial queue with the server subject to starting failures

In this paper, we study a retrial queueing model with the server subject to starting failures. We first present the necessary and sufficient condition for the system to be stable and derive analytical results for the queue length distribution as well as some performance measures of the system in steady state. We show that the general stochastic decomposition law forM/G/1 vacation models also holds for the present system. Finally, we demonstrate that a few well known queueing models are special cases of the present model and discuss various interpretations of the stochastic decomposition law when applied to each of these special cases.Partially supported by the Natural Sciences and Engineering Research Council of Canada, grant OGP0046415.Partially supported by internal research grant of Mount Saint Vincent University.     

Optimal control of an M/M/2 queueing system with finite capacity operating under the triadic (0,Q,N,M) policy

operating under the triadic (0,Q, N,M) policy, where L is the maximum number of customers in the system. The number of working servers can be adjusted one at a time at arrival epochs or at service completion epochs depending on the number of customers in the system. Analytic closed-form solutions of the controllable M/M/2 queueing system with finite capacity operating under the triadic (0,Q, N,M) policy are derived. This is a generalization of the ordinary M/M/2 and the controllable M/M/1 queueing systems in the literature. The total expected cost function per unit time is developed to obtain the optimal operating (0,Q, N,M) policy at minimum cost.     

Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System

In this paper for the M(n)/M(n)/s+GI system, i.e. for a s-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general M(n)/M/s+GI system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct.     

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.     

The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.