An M/G/1 Queueing Model with Gated Random Order of Service

We analyse an M/G/1 queueing model with gated random order of service. In this service discipline there are a waiting room, in which arriving customers are collected, and a service queue. Each time the service queue becomes empty, all customers in the waiting room are put instantaneously and in random order into the service queue. The service times of customers are generally distributed with finite mean. We derive various bivariate steady-state probabilities and the bivariate Laplace–Stieltjes transform (LST) of the joint distribution of the sojourn times in the waiting room and the service queue. The derivation follows the line of reasoning of Avi-Itzhak and Halfin [4]. As a by-product, we obtain the joint sojourn times LST for several other gated service disciplines.     

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.     

Analyzing the discrete-timeG (G)/Geo/1 queue using complex contour integration

In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as theG ^(G)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases we rediscover a number of well-known results, such as Hunter's result for theG/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more generalG ^(G)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.Both authors wish to thank the Belgian National Fund for Scientific Research (NFWO) for support of this work.     

The single-server queue with uniformly limited actual waiting times

This paper studies a single-server queueing system in which no customer has to wait for a duration longer than a constant K. If the waiting time is longer than K, then the service time of the previous customer will have to be cut short. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary solutions for the system.     

Efficient computational analysis of non-exhaustive service vacation queues: BMAP/R/1/N(∞) under gated-limited discipline

This paper analyzes the finite-buffer single server queue with vacation(s). It is assumed that the arrivals follow a batch Markovian arrival process (BMAP) and the server serves customers according to a non-exhaustive type gated-limited service discipline. It has been also considered that the service and vacation distributions possess rational Laplace-Stieltjes transformation (LST) as these types of distributions may approximate many other distributions appeared in queueing literature. Among several batch acceptance/rejection strategies, the partial batch acceptance strategy is discussed in this paper. The service limit L (1 ≤ L ≤ N) is considered to be fixed, where N is the buffer-capacity excluding the one in service. It is assumed that in each busy period the server continues to serve until either L customers out of those that were waiting at the start of the busy period are served or the queue empties, whichever occurs first. The queue-length distribution at vacation termination/service completion epochs is determined by solving a set of linear simultaneous equations. The successive substitution method is used in the steady-state equations embedded at vacation termination/service completion epochs. The distribution of the queue-length at an arbitrary epoch has been obtained using the supplementary variable technique. The queue-length distributions at pre-arrival and post-departure epoch are also obtained. The results of the corresponding infinite-buffer queueing model have been analyzed briefly and matched with the previous model. Net profit function per unit of time is derived and an optimal service limit and buffer-capacity are obtained from a maximal expected profit. Some numerical results are presented in tabular and graphical forms.     

Single-Server Queue with Markov-Dependent Inter-Arrival and Service Times

In this paper we study a single-server queue where the inter-arrival times and the service times depend on a common discrete time Markov chain. This model generalizes the well-known MAP/G/1 queue by allowing dependencies between inter-arrival and service times. The waiting time process is directly analyzed by solving Lindley's equation by transform methods. The Laplace–Stieltjes transforms (LST) of the steady-state waiting time and queue length distribution are both derived, and used to obtain recursive equations for the calculation of the moments. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correlation between the inter-arrival and service times. An erratum to this article is available at .     

Assembly-like queues with finite capacity: Bounds,asymptotics and approximations

In this paper we study a queueing model of assembly-like manufacturing operations. This study was motivated by an examination of a circuit pack testing procedure in an AT & T factory. However, the model may be representative of many manufacturing assembly operations. We assume that customers fromn classes arrive according to independent Poisson processes with the same arrival rate into a single-server queueing station where the service times are exponentially distributed. The service discipline requires that service be rendered simultaneously to a group of customers consisting of exactly one member from each class. The server is idle if there are not enough customers to form a group. There is a separate waiting area for customers belonging to the same class and the size of the waiting area is the same for all classes. Customers who arrive to find the waiting area for their class full, are lost. Performance measures of interest include blocking probability, throughput, mean queue length and mean sojourn time. Since the state space for this queueing system could be large, exact answers for even reasonable values of the parameters may not be easy to obtain. We have therefore focused on two approaches. First, we find upper and lower bounds for the mean sojourn time. From these bounds we obtain the asymptotic solutions as the arrival rate (waiting room, service rate) approaches zero (infinity). Second, for moderate values of these parameters we suggest an approximate solution method. We compare the results of our approximation against simulation results and report good correspondence.     

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.     

Waiting time approximation in single-class queueing systems with multiple types of interruptions: modeling congestion at waterways entrances

We consider a single-server queue subject to multiple types of operation-independent interruptions motivated by operations and vessel queueing at entrances of waterways. A case in point is the Strait of Istanbul. We are using waiting-time arguments and service completion time analysis to obtain the expected waiting time of a customer (vessel) in the aforementioned queue with single-class of customers and k non-simultaneous and possibly simultaneous service interruptions. In the analysis, we have used arguments and assumptions from the Strait of Istanbul that are also valid for narrow waterways at large.     

The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).     

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).     

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N  -policy GI/Geo/1$\mathit{GI}/\mathit{Geo}/1$ queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.     

A tollbooth tandem queue with heterogeneous servers

We study a tandem queueing system with K servers and no waiting space in between. A customer needs service from one server but can leave the system only if all down-stream servers are unoccupied. Such a system is often observed in toll collection during rush hours in transportation networks, and we call it a tollbooth tandem queue. We apply matrix-analytic methods to study this queueing system, and obtain explicit results for various performance measures. Using these results, we can efficiently compute the mean and variance of the queue lengths, waiting time, sojourn time, and departure delays. Numerical examples are presented to gain insights into the performance and design of the tollbooth tandem queue. In particular, it reveals that the intuitive result of arranging servers in decreasing order of service speed (i.e., arrange faster servers at downstream stations) is not always optimal for minimizing the mean queue length or mean waiting time.     

Analysis of a Two‐Phase Queueing System with Vacations and Bernoulli Feedback

Abstract

We consider a two‐phase queueing system with server vacations and Bernoulli feedback. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. Each customer who completes the individual service returns to the tail of the second phase service queue with probability 1 ? σ. If the system becomes empty at the moment of the completion of the second phase services, the server takes vacations until he finds customers. This type of queueing problem can be easily found in computer and telecommunication systems. By deriving a relationship between the generating functions for the system size at various embedded epochs, we obtain the system size distribution at an arbitrary time. The exhaustive and gated cases for the batch service are considered.     

The Analysis of a General Input Queue with N Policy and Exponential Vacations

This paper studies a single removable server in a G/M/1 queueing system with finite capacity where the server applies an N policy and takes multiple vacations when the system is empty. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential and deterministic interarrival time distributions. We establish the distributions of the number of customers in the queue at pre-arrival epochs and at arbitrary epochs, as well as the distributions of the waiting time and the busy period.     

Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server

We study a single removable server in an M/G/1 queueing system operating under the N policy in steady-state. The server may be turned on at arrival epochs or off at departure epochs. Using the maximum entropy principle with several well-known constraints, we develop the approximate formulae for the probability distributions of the number of customers and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with exact analytic results for three different service time distributions, exponential, 2-stage Erlang, and 2-stage hyper-exponential. The maximum entropy approximation approach is accurate enough for practical purposes. We demonstrate, through the maximum entropy principle results, that the N policy M/G/1 queueing system is sufficiently robust to the variations of service time distribution functions.     

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.     

Workload and Waiting Time Analyses of MAP/G/1 Queue under D-policy

We study the workload (unfinished work) and the waiting time of the queueing system with MAP arrivals under D-policy. The D-policy stipulates that the idle server begin to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We first set up the system equations for workload and obtain the steady-state distributions of workloads at an arbitrary idle and busy points of time. We then proceed to obtain the waiting time distribution of an arbitrary customer based on the workload results. The M/G/1/D-policy queue will be investigated as a special case.