Optimal control of noncollaborative servers in two‐stage tandem queueing systems

Abstract

We consider two‐stage tandem queueing systems with dedicated servers in each station and a flexible server that is trained to serve both stations. We assume no arrivals, exponential service times, and linear holding costs for jobs present in the system. We study the optimal dynamic assignment of servers to jobs assuming a noncollaborative work discipline with idling and preemptions allowed. For larger holding costs in the first station, we show that (i) nonidling policies are optimal and (ii) if the flexible server is not faster than the dedicated servers, the optimal server allocation strategy has a threshold‐type structure. For all other cases, we provide numerical results that support the optimality of threshold‐type policies. Our numerical experiments also indicate that when the flexible server is faster than the dedicated server of the second station, the optimal policy may have counterintuitive properties, which is not the case when a collaborative service discipline is assumed. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 435–446, 2014

Optimal Dynamic Allocation of Collaborative Servers in Two Station Tandem Systems

“…Lemma 1 states that the minimum expected cost increases with the number of jobs in the system. It can be proved by induction on the number of jobs (as in [15]) or by straightforward sample path arguments.…”

“…The necessity of the condition is proved in the next section (Theorem 4). The sufficiency is a consequence of the following lemma, whose proof can be obtained by induction on the number of jobs in Station 2 (see [15]) or by sample path arguments.…”

“…Moreover, a nonidling discipline for at least the dedicated servers was assumed. Both of these conditions were relaxed by Pandelis [15], who provided conditions under which non-idling policies that are characterized by a single switching curve are optimal for clearing systems under a non-collaborative service discipline.…”

Optimization of flexible production and supply systems

“…Moreover, a non-idling discipline for at least the dedicated servers was assumed. Both of these conditions were relaxed by Pandelis [59]. For clearing systems he showed that non-idling policies are optimal when the holding cost rate in the upstream station is not less than the corresponding rate in the downstream station, and for this case he provided conditions on service rates that ensure that the optimal server allocation is characterized by a single switching curve under a non-collaborative service discipline.…”