Abstract
This paper analyzes customers’ impatience in Markovian queueing system with multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. During the working vacation period, if there are customers in the queue, the vacation can be interrupted at a service completion instant and the server begins a regular busy period with probability 1-q or continues the vacation with probability q. We obtain the probability generating functions of the stationary state probabilities and deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures such as the mean system size, the proportion of customers served, the rate of abandonment due to impatience, and the mean sojourn time of a customer served are derived. We obtain the stochastic decomposition structures of the queue length and waiting time. Finally, some numerical results to show the impact of model parameters on performance measures of the system are presented.
Highlights
Queueing systems with customers’ impatience and server vacations have been analyzed due to their wide applications in real-life congestion problems such as computer and communication systems, telecommunication systems, traffic systems, service systems, and manufacturing and production systems
If ρ < 1, the stationary queue length L can be decomposed into the sum of two independent random variables: L = Lc + Ld, where Lc is the stationary queue length of a classical M/M/1 queue without vacations and the additional stationary queue length Ld due to the Bernoulli schedule vacation interruption has a distribution with its pgf: Ld (z)
If ρ < 1, the stationary waiting time W can be decomposed into the sum of two independent random variables: W = Wc + Wd, where Wc is the stationary waiting time of a M/M/1 queue without vacations, which has an exponential distribution with the parameter μ(1 − ρ), and the additional stationary waiting time Wd due to the Bernoulli schedule vacation interruption has a distribution with its Laplace Stieltjes transform (LST): Wd∗
Summary
Queueing systems with customers’ impatience and server vacations have been analyzed due to their wide applications in real-life congestion problems such as computer and communication systems, telecommunication systems, traffic systems, service systems, and manufacturing and production systems. We derive an analytic expression for the stationary distribution of probability generating functions of the number of customers in the system when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Some performance measures such as mean system size, the proportion of customers served, the rate of abandonment due to impatience, and the mean sojourn time of a customer served have been evaluated.
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