Analysis of Impatient Customers in Queues with Bernoulli Schedule Working Vacations and Vacation Interruption

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 or continues the vacation with probability . 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. 1. Introduction 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. Impatience is the most noted feature as individuals always feel worried and impatient throughout when waiting for service which leads to loss of potential customers. It has special implication for the business world as it has an adverse impact on the revenue generation of a business organization. The customer’s impatient behavior should be needed in the study of queueing system to model real conditions precisely. Occasional operation of a service may be economically invoking when entire time service would result in substantial server idle time or would prevent the utilization of the server in different productive capacities. On the other hand, the server remaining not working for periods of time might gain the probability of customer losses due to balking and reneging. Vacation queues have been examined extensively in Ke et al. [1], Tian and Zhang [2], and so forth. Yue et al. [3] studied an queueing system with balking, reneging, and server vacations. There are several situations where the server stays active during the vacation period. The server can provide service at a lower speed during the working vacation (WV) period instead of stopping service