scholarly journals A Batch Arrival Single Server Queue with Server Providing General Service in Two Fluctuating Modes and Reneging during Vacation and Breakdowns

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Monita Baruah ◽  
Kailash C. Madan ◽  
Tillal Eldabi
Keyword(s):  
Repair Process ◽  
Batch Arrival ◽  
Single Server ◽  
Single Server Queue ◽  
Numerical Illustration ◽  
Random Breakdown ◽  
Special Cases ◽  
Server Queue ◽  
Service Rates ◽  
General Service

We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results.

Some perturbation results for the single-server queue with Poisson input

1965 ◽  
Vol 2 (2) ◽  
pp. 462-466 ◽  
Author(s):  
A. M. Hasofer
Keyword(s):  
Service Time ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Image Position ◽  
General Service

In a previous paper [2] the author has studied the single-server queue with non-homogeneous Poisson input and general service time, with particular emphasis on the case when the parameter of the Poisson input is of the form

A single-server queue with server vacations and a class of non-renewal arrival processes

1990 ◽  
Vol 22 (3) ◽  
pp. 676-705 ◽  
Author(s):  
David M. Lucantoni ◽  
Kathleen S. Meier-Hellstern ◽  
Marcel F. Neuts
Keyword(s):  
Waiting Times ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Arbitrary Time ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Server Queue ◽  
Arrival Processes ◽  
Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

scholarly journals Estimating the Probability of Ruin for Variable Premiums by Simulation

1996 ◽  
Vol 26 (1) ◽  
pp. 93-105 ◽  
Author(s):  
Frédéric Michaud
Keyword(s):  
Sample Path ◽  
Risk Theory ◽  
Single Server ◽  
Single Server Queue ◽  
Storage Process ◽  
Probability Of Ruin ◽  
Surplus Process ◽  
Special Cases ◽  
Server Queue ◽  
Waiting Time Process

AbstractThere is a duality between the surplus process of classical risk theory and the single-server queue. It follows that the probability of ruin can be retrieved from a single sample path of the waiting time process of the single-server queue. In this paper, premiums are allowed to vary. It has been shown that the stationary distribution of a corresponding storage process is equal to the survival probability (with variable premiums). Thus by simulation of the corresponding storage process, the probability of ruin can be obtained. The special cases where the surplus earns interest and the premiums are charged by layers are considered and illustrated numerically.

Waiting time and workload in queues with periodic Poisson input

1989 ◽  
Vol 26 (02) ◽  
pp. 390-397 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Differential Equation ◽  
Waiting Time ◽  
Arrival Rate ◽  
Time Of Day ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Service Distribution ◽  
Server Queue ◽  
General Service

This paper develops moment formulas for asymptotic workload and waiting time in a single-server queue with periodic Poisson input and general service distribution. These formulas involve the corresponding moments of waiting-time (workload) for the M/G/1 system with the same average arrival rate and service distribution. In certain cases, all the terms in the formulas can be computed exactly, including moments of workload at each ‘time of day.' The approach makes use of an asymptotic version of the Takács [12] integro-differential equation, together with representation results of Harrison and Lemoine [3] and Lemoine [6].

scholarly journals The transient behaviour of the queueing system Gi/M/1

1963 ◽  
Vol 3 (2) ◽  
pp. 249-256 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Distribution Function ◽  
Explicit Expression ◽  
Queueing System ◽  
Single Server ◽  
Single Server Queue ◽  
Transient Behaviour ◽  
Special Cases ◽  
Server Queue ◽  
Interarrival Times ◽  
Lagrange Series

SUMMARYWe consider a single server queue for which the interarrival times are identically and independently distributed with distribution function A(x) and whose service times are distributed independently of each other and of the interarrival times with distribution function B(x) = 1 − e−x, x ≧ 0. We suppose that the system starts from emptiness and use the results of P. D. Finch [2] to derive an explicit expression for qnj, the probability that the (n + 1)th arrival finds more than j customers in the system. The special cases M/M/1 and D/M/1 are considerend and it is shown in the general case that qnj is a partial sum of the usual Lagrange series for the limiting probability .

On the single-server queue with non-homogeneous Poisson input and general service time

1964 ◽  
Vol 1 (2) ◽  
pp. 369-384 ◽  
Author(s):  
A. M. Hasofer
Keyword(s):  
Periodic Function ◽  
Waiting Time ◽  
Service Time ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Image Position ◽  
General Service

In this paper, a single-server queue with non-homogeneous Poisson input and general service time is considered. Particular attention is given to the case where the parameter of the Poisson input λ(t) is a periodic function of the time. The approach is an extension of the work of Takács and Reich . The main result of the investigation is that under certain conditions on the distribution of the service time, the form of the function λ(t) and the distribution of the waiting time at t = 0, the probability of a server being idle P0 and the Laplace transform Ω of the waiting time are both asymptotically periodic in t. Putting where b(t) is a periodic function of time, it is shown that both Po and Ω can be expanded in a power series in z, and a method for calculating explicitly the asymptotic values of the leading terms is obtained.In many practical queueing problems, it is expected that the probability of arrivals will vary periodically. For example, in restaurants or at servicestations arrivals are more probable at rush hours than at slack periods, and rush hours are repeated day after day

Some perturbation results for the single-server queue with Poisson input

1965 ◽  
Vol 2 (02) ◽  
pp. 462-466
Author(s):  
A. M. Hasofer
Keyword(s):  
Service Time ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Image Position ◽  
General Service

In a previous paper [2] the author has studied the single-server queue with non-homogeneous Poisson input and general service time, with particular emphasis on the case when the parameter of the Poisson input is of the form

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