The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1

1967 ◽  
Vol 4 (01) ◽  
pp. 162-179 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Busy Period ◽  
Transition Probabilities ◽  
Queueing Systems ◽  
Entrance Time ◽  
Distribution Of The Maximum ◽  
Number Of Customers

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.

The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1

1967 ◽  
Vol 4 (1) ◽  
pp. 162-179 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Busy Period ◽  
Transition Probabilities ◽  
Queueing Systems ◽  
Entrance Time ◽  
Distribution Of The Maximum ◽  
Number Of Customers

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.

The Trivariate Distribution of the Maximum Queue Length, the Number of Customers Served and the Duration of the Busy Period for the M/G/1 Queueing System

1969 ◽  
Vol 6 (1) ◽  
pp. 154-161 ◽  
Author(s):  
E.G. Enns
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Queueing System ◽  
Single Server ◽  
List Type ◽  
Type Number ◽  
Distribution Of The Maximum ◽  
Maximum Queue Length ◽  
Number Of Customers

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:1.The duration of the busy period.2.The number of customers served during the busy period.3.The maximum number of customers in the queue during the busy period.

The Trivariate Distribution of the Maximum Queue Length, the Number of Customers Served and the Duration of the Busy Period for the M/G/1 Queueing System

1969 ◽  
Vol 6 (01) ◽  
pp. 154-161 ◽  
Author(s):  
E.G. Enns
Keyword(s):  
Queue Length ◽  
Busy Period ◽  
Queueing System ◽  
Single Server ◽  
List Type ◽  
Type Number ◽  
Distribution Of The Maximum ◽  
Maximum Queue Length ◽  
Number Of Customers

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are: 1. The duration of the busy period. 2. The number of customers served during the busy period. 3. The maximum number of customers in the queue during the busy period.

The busy period of the E k/G/1 queue by finite waiting room

1975 ◽  
Vol 7 (02) ◽  
pp. 416-430
Author(s):  
A. L. Truslove
Keyword(s):  
Markov Chain ◽  
Busy Period ◽  
Waiting Room ◽  
Queueing Process ◽  
Number Of Customers

For the E k /G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of the busy period, together with the number of customers who arrive, and the number of customers served, during that period, is obtained. The limit as the size of the waiting room becomes infinite is found.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

scholarly journals On the ergodic distribution of oscillating queueing systems

2003 ◽  
Vol 16 (4) ◽  
pp. 311-326 ◽  
Author(s):  
Mykola Bratiychuk ◽  
Andrzej Chydzinski
Keyword(s):  
Queueing Systems ◽  
Numerical Example ◽  
New Class ◽  
Ergodic Distribution ◽  
Number Of Customers ◽  
Special Case

This paper examines a new class of queueing systems and proves a theorem on the existence of the ergodic distribution of the number of customers in such a system. An ergodic distribution is computed explicitly for the special case of a G/M−M/1 system, where the interarrival distribution does not change and both service distributions are exponential. A numerical example is also given.

scholarly journals Concavity of the throughput of tandem queueing systems with finite buffer storage space

1990 ◽  
Vol 22 (03) ◽  
pp. 764-767 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Sample Path ◽  
Concave Function ◽  
Time Interval ◽  
Single Server ◽  
Finite Buffer ◽  
Buffer Storage ◽  
Unlimited Supply ◽  
Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

q-SERIES IN MARKOV CHAINS WITH BINOMIAL TRANSITIONS

2008 ◽  
Vol 23 (1) ◽  
pp. 75-99 ◽  
Author(s):  
Antonis Economou ◽  
Stella Kapodistria
Keyword(s):  
Busy Period ◽  
Hypergeometric Series ◽  
Binomial Coefficients ◽  
Single Server ◽  
Transition Rates ◽  
Extensive Analysis ◽  
Service Completion ◽  
Markovian Queue ◽  
Number Of Customers ◽  
Sojourn Time Distributions

We consider a single-server Markovian queue with synchronized services and setup times. The customers arrive according to a Poisson process and are served simultaneously. The service times are independent and exponentially distributed. At a service completion epoch, every customer remains satisfied with probability p (independently of the others) and departs from the system; otherwise, he stays for a new service. Moreover, the server takes multiple vacations whenever the system is empty.Some of the transition rates of the underlying two-dimensional Markov chain involve binomial coefficients dependent on the number of customers. Indeed, at each service completion epoch, the number of customers n is reduced according to a binomial (n, p) distribution. We show that the model can be efficiently studied using the framework of q-hypergeometric series and we carry out an extensive analysis including the stationary, the busy period, and the sojourn time distributions. Exact formulas and numerical results show the effect of the level of synchronization to the performance of such systems.

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