Time-dependent analysis of M/G/1 vacation models with exhaustive service

We analyze the time-dependent process in severalM/G/1 vacation models, and explicitly obtain the Laplace transform (with respect to an arbitrary point in time) of the joint distribution of server state, queue size, and elapsed time in that state. Exhaustive-serviceM/G/1 systems with multiple vacations, single vacations, an exceptional service time for the first customer in each busy period, and a combination ofN-policy and setup times are considered. The decomposition property in the steady-state joint distribution of the queue size and the remaining service time is demonstrated.     

An M/G/1 queue with second optional service

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate (>0) all demand the first essential service, whereas only some of them demand the second optional service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/₂ (₂>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/E_k/1 and M/M/1 have been derived as particular cases.     

AN M/G/1 RETRIAL QUEUE WITH SECOND MULTI-OPTIONAL SERVICE, FEEDBACK AND UNRELIABLE SERVER

An M/G/1 retrial queue with two-phase service and feedback is studied in this paper, where the server is subject to starting failures and breakdowns during service. Primary customers get in the system according to a Poisson process, and they will receive service immediately if the server is available upon arrival. Otherwise, they will enter a retrial orbit and are queued in the orbit in accordance with a first-come-first-served （FCFS） discipline. Customers are allowed to balk and renege at particular times. All customers demand the first “essential” service, whereas only some of them demand the second “multi-optional” service. It is assumed that the retrial time, service time and repair time of the server are all arbitrarily distributed. The necessary and sufficient condition for the system stability is derived. Using a supplementary variable method, the steady-state solutions for some queueing and reliability measures of the system are obtained.     

An M/G/1-type queuing model with service times depending on queue length

A study is made of an M/G/1-type queuing model in which customers receive one type of service until such time as, at the end of a service, the queue size is found to exceed a given value N, N ≥ 1. Then a second type of service is put into effect and remains in use until the queue size is reduced to a fixed value K, 0 ≤ K ≤ N. Equations are derived for the stationary probabilities both at departure times and at general times. An algorithm is developed that allows the rapid computation of the mean queue length and some important probabilities.     

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.     

Approximations for the conditional waiting times in the GI/G/c queue

This note presents a two-moment approximation for the conditional average waiting time in the standard multi-server queue and an approximation for the tail probabilities of the conditional waiting time distribution in the standard single-server queue. These approximations have been tested by extensive numerical experiments.     

Optimality of a Threshold Policy in theM/M/ queue with repeated vacations

Consider anM/M/1 queueing system with server vacations where the server is turned off as soon as the queue gets empty. We assume that the vacation durations form a sequence of i.i.d. random variables with exponential distribution. At the end of a vacation period, the server may either be turned on if the queue is non empty or take another vacation. The following costs are incurred: a holding cost ofh per unit of time and per customer in the system and a fixed cost of each time the server is turned on. We show that there exists a threshold policy that minimizes the long-run average cost criterion. The approach we use was first proposed in Blanc et al. (1990) and enables us to determine explicitly the optimal threshold and the optimal long-run average cost in terms of the model parameters.     

The M/G/1 retrial queue with bernoulli schedule

We consider anM/G/1 retrial queue with infinite waiting space in which arriving customers who find the server busy join either (a) the retrial group with probabilityp in order to seek service again after a random amount of time, or (b) the infinite waiting space with probabilityq(=1–p) where they wait to be served. The joint generating function of the numbers of customers in the two groups is derived by using the supplementary variable method. It is shown that our results are consistent with known results whenp=0 orp=1.     

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.     

VIRTUAL WAITING TIMES IN PRIORITY-M/G/1 QUEUES WITH VACARIONS

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Distribution of spatial requirements for a MAP/G/1 queue when space and service times are dependent

In this paper, we consider a MAP/G/1 queue in which each customer arrives with a service and a space requirement, which could be dependent. However, the space and service requirements of different customers are assumed to be independent. Each customer occupies its space requirement in a buffer until it has completely received its service, at which time, it relinquishes the space it occupied. We study and solve the problem of finding the steady-state distribution of the total space requirement of all customers present in the system. In the process of doing so, we also generalize the solution of the MAP/G/1 queue and find the time-average joint distribution of the queue-length, the state of the arrival process and the elapsed service time, conditioned on the server being busy. This problem has applications to the design of buffer requirements for a computer or communication system.     

Time-dependent analysis of a queue with batch arrivals andn levels of nonpreemptive priority

A queueing system with batch arrivals andn classes of customers with nonpreemptive priorities between them is considered. Each batch arrives according to the Poisson distribution and contains customers of all classes while the service times follow arbitrary distributions with different probability density functions for each class. For such a model the system states probabilities both in the transient and in the steady state are analysed and also expressions for the Laplace transforms of the busy period densities for each class and for the general busy period are obtained.     

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.     

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.     

Application of the method of collective marks to someM/G/1 vacation models with exhaustive service

We use the Method of Collective Marks to analyze some time-dependent processes in theM/G/1 queue with single and multiple vacations. With the server state specified at a fixed timet>0, the Laplace transforms with respect tot of mixed transforms for the joint distribution of the number of departures by timet, the queue length, the virtual waiting time, the elapsed and remaining service/vacation times at timet are derived by means of probabilistic interpretations. The Laplace-Stieltjes transform of the virtual waiting time at timet is also given. Some well known results are special cases.This research was supported by the University of Amsterdam.     

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.     

Scheduling in a multi-class series of queues with deterministic service times

We consider a problem of scheduling in a multi-class network of single-server queues in series, in which service times at the nodes are constant and equal. Such a model has potential application to automated manufacturing systems or packet-switched communication networks, where a message is divided into packets (or cells) of fixed lengths. The network is a series-type assembly or transfer line, with the exception that there is an additional class of jobs that requires processing only at the first node (class 0). There is a holding cost per unit time that is proportional to the total number of customers in the system. The objective is to minimize the (expected) total discounted holding cost over a finite or an infinite horizon. We show that an optimal policy gives priority to class-0 jobs at node 1 when at least one of a set ofm–1 inequalities on partial sums of the components of the state vector is satisfied. We solve the problem by two methods. The first involves formulating the problem as a (discrete-time) Markov decision process and using induction on the horizon length. The second is a sample-path approach using an interchange argument to establish optimality.The research of this author was supported by the National Science Foundation under Grant No. DDM-8719825. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.     

Tail probabilities of low-priority waiting times and queue lengths in MAP/GI/1 queues

We consider the problem of estimating tail probabilities of waiting times in statistical multiplexing systems with two classes of sources – one with high priority and the other with low priority. The priority discipline is assumed to be nonpreemptive. Exact expressions for the transforms of these quantities are derived assuming that packet or cell streams are generated by Markovian Arrival Processes (MAPs). Then a numerical investigation of the large-buffer asymptotic behavior of the the waiting-time distribution for low-priority sources shows that these asymptotics are often non-exponential.     

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.     

M/G/1/K queues withN-policy and setup times

A steady-state analysis is given for M/G/1/K queues with combinedN-policy and setup times before service periods. The queue length distributions and the mean waiting times are obtained for the exhaustive service system, the gated service system, the E-limited service system, and the G-limited service system. Numerical examples are also provided.