Probability Density Function of M/G/1 Queues Under (0,K) Control Policies: A Special Case (original) (raw)
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For the M/G/1/m queue with the forced vacations in work after n customers served in succession probabilities of states of limiting steady-state process are found. 1. Introduction. In case of arbitraryly distributed service time and exponentially distributed interarrival time existence of limiting steady-state process for the single-server queueing system with unlimited queue is proved by a method of embedded Markov chains [ 1, p. 98 ]. For M/G/1/m queue probabilities of states of limiting steady-state process are found in [ 2, p. 235 ]. Below we study M/G/1/m queue which feature consists that after ï customers served in succession the forced vacations in work of system occur. Necessity of such pause can be caused by technological or other reasons. For example, the technical device which consistently carries out homogeneous operations, at designing can be calculated on the limited number of operations which are carried out without interruption between operations.
Analytic and numerical aspects of batch service queues with single vacation
Computers & Operations Research, 2005
This paper deals with an M=G=1 batch service queue where customers are served in batches of maximum size b with a minimum threshold value a. The server takes a single vacation when he ÿnds less than a customers after the service completion. The vacation time of the server is arbitrarily distributed. Using the supplementary variable method we obtain the probability generating functions of the queue length distributions at various epochs. We also obtain relations among queue length distributions at arbitrary, service (vacation) termination epochs. Further their evaluation is also discussed. Finally, some numerical results and graphs are presented.
Computing queue length distributions in MAP/G/1/N queue under single and multiple vacation
Applied Mathematics and Computation, 2006
This paper studies a single server queue with finite waiting room in which the server takes vacation(s) whenever the system becomes empty and we consider both single and multiple vacation(s). Whereas the input process is a Markovian Arrival Process (MAP), the service and vacation times are arbitrarily distributed. The distributions of number of customers in the queue at service completion, vacation termination, departure, arbitrary and pre-arrival epochs have been obtained. Computational procedure has been given when the service-and vacation-time distributions are of phase type (PH-distribution).
Analysis and Computational Algorithm for Queues with State-Dependent Vacations I: G/M(n)/1/K
Journal of Systems Science and Complexity, 2006
In this paper we study a queueing system with state-dependent services and state-dependent vacations, or simply G/M (n)/1/K. Since the service rate is state-dependent, this system includes G/M/c and G/M/c/K queues with various types of station vacations as special cases. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. The only input requirement is the Laplace-Stieltjes transform of the interarrival distribution as well as the state-dependent service rate and state-dependent vacation rate. In a subsequent companion paper, we study its dual system M (n)/G/1/K queue with statedependent vacations.
Stationary queue and server content distribution of a vacation queue with N -policy and set up time
Authorea (Authorea), 2023
Due to unambiguous applications of bulk service vacation queues with set up time in various fields such as pharmaceutical, aerospace engineering, automobile industries etc., in this article, we analyze an infinite-buffer batch-size-dependent bulk service queue with single and multiple vacations, N-policy and set up time. Customers/packets/units are served by a single server according to general bulk service (a,b) rule. The service time of a batch dynamically vary with the batch-size and follows the general type of distribution which covers a large scale of distributions. Firstly, we generate the steady-state system equations. The main intend of this study is to obtain the complete joint distribution of queue-length and server content at service completion epoch, for which the bivariate probability generating function has been derived. We extract the joint distribution which is presented in a quite simple form and using those we find the joint distribution at arbitrary epoch beside some marginal distributions and performance measures. Finally, several numerical examples along with graphical sketches have been provided to verify the analytical results and to provide inner feeling to the system designers.
Journal of Applied Mathematics and Stochastic Analysis, 2005
We consider a finite-buffer single-server queue with Markovian arrival process (MAP) where the server serves a limited number of customers, and when the limit is reached it goes on vacation. Both single-and multiple-vacation policies are analyzed and the queue length distributions at various epochs, such as pre-arrival, arbitrary, departure, have been obtained. The effect of certain model parameters on some important performance measures, like probability of loss, mean queue lengths, mean waiting time, is discussed. The model can be applied in computer communication and networking, for example, performance analysis of token passing ring of LAN and SVC (switched virtual connection) of ATM.
Steady State Analysis of an M/D/1 Queue with Coxian-2 Server Vacations and a Single Vacation Policy
International Journal of Information and Management Sciences
We analyze the steady state behavior of an M/D/1 queue with Bernoulli shedules and Coxian-2 server vacations. Customers arrive one by one at the system in a Poisson stream. The service time of a customer is assumed to be deterministic. At each service completion epoch, the server may opt to take a vacation with probability p or else with probability 1-p may continue to be available in the system for the next service. The vacation period of the server is assumed to have a Coxian-2 distribution. We obtain steady state probability generating functions for the queue size and the system size in explicit and closed forms. Some particular cases of interst including known results of the M/D/1 queue have been derived.
A Single Server Queue with Coxian-2 Service and One-Phase Vacation (M/C-2/M/1 Queue)
Open Journal of Applied Sciences, 2021
In this paper, we study a single server queueing system with Coxian-2 service. In Particular, we study M/C-2/M/1 queue with Coxian-2 service and exponential vacation. We assume that units (customers) arrive at the system one by one in a Poisson process and the server provides one-by-one service based on first in first out (FIFO) rule. We obtained the steady state queue size distributions in terms of the probability generating functions, the average number of customers and their average waiting time in the system as well as in the queue.
Computers & Industrial Engineering, 2013
This paper is concerned with the analysis of a single server queueing system subject to Bernoulli vacation schedules with server setup and close down periods. An explicit expression for the probability generating function of the number of customers present in the system is obtained by using imbedded Markov chain technique. The steady state probabilities of no customer in the system at the end of vacation termination epoch and a service completion epoch are derived. The mean number of customers served during a service period and the mean number of customers in the system at an arbitrary epoch are investigated under steady state. Further, the Laplace-Stieltjes transform of the waiting time distribution and its corresponding mean are studied. Numerical results are provided to illustrate the effect of system parameters on the performance measures.
Quality Technology & Quantitative Management, 2021
Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacation where customers/messages arrive according to the Bernoulii process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at arbitrary slot. We also provide several marginal distributions and performance measures for the utilization of the vendor. Transmission of data through a particular channel is skipped due to the high transmission error. As the discrete phase type distribution plays a noteworthy role to control this error, we include numerical example where service time distribution follows discrete phase type distribution. A comparison between batch-size dependent and independent service has been drawn through the graphical representation of some performance measures and total system cost.