International Journal of Computational and Applied Mathematics

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Time Dependent Solution of Batch Arrival Queueing Systemwith Random Breakdowns and Second Optional Server Vacation


Affiliations
1 Department of Mathematics, Pondicherry Engineering College, Pondicherry, India
2 Department of Mathematics, Arunai Engineering College, Thiruvannamalai, India
     

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This paper investigates a single server queue with Poisson arrivals, general (arbitrary) service time distributions and Bernoulli vacation subject to random breakdowns. However, after the completion of a service, the server will take Bernoulli vacation, that the server make take a vacation with probability θ or may continue to stay in the system with probability 1 − θ for serving the next customer, if any. In addition to this, the vacation period of the server has two phases in which first phase is compulsory followed by the second phase in a such way a that the server may choose second phase with probability p or may return back to the system with probability 1−p and the vacation time follows general (arbitrary) distribution. The system may breakdown at random with mean break down rate α and repair process starts immediately in which the repair time follows exponential distribution with mean repair rate β. We obtain the time dependent probability generating functions in terms of their Laplace transforms and the corresponding steady state results explicitly. Also we derive the average number of customers in the queue and the average waiting time in closed form.

AMS subject classification: 60K25, 60K30


Keywords

M[X]/G/1 Queue, Bernoulli Vacation, Probability Generating Function, Transient State, Idle State, Steady State, Mean Queue Length, Mean Waiting Time
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  • Time Dependent Solution of Batch Arrival Queueing Systemwith Random Breakdowns and Second Optional Server Vacation

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Authors

G. Ayyappan
Department of Mathematics, Pondicherry Engineering College, Pondicherry, India
S. Shyamala
Department of Mathematics, Arunai Engineering College, Thiruvannamalai, India

Abstract


This paper investigates a single server queue with Poisson arrivals, general (arbitrary) service time distributions and Bernoulli vacation subject to random breakdowns. However, after the completion of a service, the server will take Bernoulli vacation, that the server make take a vacation with probability θ or may continue to stay in the system with probability 1 − θ for serving the next customer, if any. In addition to this, the vacation period of the server has two phases in which first phase is compulsory followed by the second phase in a such way a that the server may choose second phase with probability p or may return back to the system with probability 1−p and the vacation time follows general (arbitrary) distribution. The system may breakdown at random with mean break down rate α and repair process starts immediately in which the repair time follows exponential distribution with mean repair rate β. We obtain the time dependent probability generating functions in terms of their Laplace transforms and the corresponding steady state results explicitly. Also we derive the average number of customers in the queue and the average waiting time in closed form.

AMS subject classification: 60K25, 60K30


Keywords


M[X]/G/1 Queue, Bernoulli Vacation, Probability Generating Function, Transient State, Idle State, Steady State, Mean Queue Length, Mean Waiting Time

References