The effect of workers with different capabilities on customer delay

Many service facilities operate seven days per week. The operations managers of these facilities face the problem of allocating personnel of varying skills and work speed to satisfy the demand for services. Furthermore, for practical reasons, periodic staffing schedule is implemented regularly. We introduce a novel approach for modeling periodic staffing schedule and analyzing the impact of employee variability on customer delay. The problem is formulated as a multiple server vacation queueing system with Bernoulli feedback of customers. At any point in time, at most one server can serve the customers. Each server incur a durations of set-up time before they can serve the customers. The customer service time and server set-up time may depend on the server. The service process is unreliable in the sense that it is possible for the customer at service completion to rejoin the queue and request for more service. The customer arrival process is assumed to satisfy a linear-quadratic model of uncertainty. We will present transient and steady-state analysis on the queueing model. The transient analysis provides a stability condition for the system to reach steady state. The steady-state analysis provides explicit expressions for several performance measures of the system. For the special case of MX/G/1 vacation queue with a gated or exhaustive service policy and Bernoulli feedback, our result reduces to a previously known result. Lastly, we show that a variant of our periodic staffing schedule model can be used to analyze queues with permanent customers. For the special case of M/G/1 queue with permanent customers and Bernoulli feedback of ordinary customers, we obtain results previously given by Boxma and Cohen (IEEE J. Select. Areas Commun. 9 (1991) 179) and van den Berg (Sojourn Times in Feedback and Processor Sharing Queues, CWI Tracts, vol. 97, Amsterdam, Netherlands, 1993).