Optimal pricing for tandem queues with finite buffers

Abstract

We consider optimal pricing for a two-station tandem queueing system with finite buffers, communication blocking, and price-sensitive customers whose arrivals form a homogeneous Poisson process. The service provider quotes prices to incoming customers using either a static or dynamic pricing scheme. There may also be a holding cost for each customer in the system. The objective is to maximize either the discounted profit over an infinite planning horizon or the long-run average profit of the provider. We show that there exists an optimal dynamic policy that exhibits a monotone structure, in which the quoted price is non-decreasing in the queue length at either station and is non-increasing if a customer moves from station 1 to 2, for both the discounted and long-run average problems under certain conditions on the holding costs. We then focus on the long-run average problem and show that the optimal static policy performs as well as the optimal dynamic policy when the buffer size at station 1 becomes large, there are no holding costs, and the arrival rate is either small or large. We learn from numerical results that for systems with small arrival rates and no holding cost, the optimal static policy produces a gain quite close to the optimal gain even when the buffer at station 1 is small. On the other hand, for systems with arrival rates that are not small, there are cases where the optimal dynamic policy performs much better than the optimal static policy.

Revenue maximization in two‐station tandem queueing systems

“…The service provider quotes a price from to customers, where and we assume that a K is the only price in that blocks all customers from entering the system (cf. Wang, Andradóttir, & Ayhan, ), unless otherwise stated. As in Çil, Karaesmen, and Örmeci (), Wang et al (), Yoon and Lewis () and Ziya, Ayhan, and Foley (), we consider that an arriving customer enters the system if the quoted price is not greater than her reservation price.…”

“…Wang, Andradóttir, & Ayhan, ), unless otherwise stated. As in Çil, Karaesmen, and Örmeci (), Wang et al (), Yoon and Lewis () and Ziya, Ayhan, and Foley (), we consider that an arriving customer enters the system if the quoted price is not greater than her reservation price. Customers who find the buffer before station 1 full upon their arrivals are lost.…”

“…Note that we primarily consider systems where arriving customers are price‐sensitive but not congestion‐sensitive in this work. Such systems have been widely considered in the literature such as Çil et al (), Wang et al (), Yoon an Lewis () and Ziya et al (). More importantly, the systems can find real‐world applications.…”

“…Moreover, our work characterizes the monotone structure of optimal policies and applies the structural results in algorithm design, while the monotone structure is neither provided nor used in algorithm design in Xia and Chen (). Wang et al () focused on the monotone structure of optimal dynamic pricing policies and the performance gap between optimal dynamic and static pricing policies for two‐station tandem queues with finite buffers. Our paper extends the work of Wang et al () toward two directions: (a) Wang et al () does not offer any computational algorithms.…”

“…Wang et al () focused on the monotone structure of optimal dynamic pricing policies and the performance gap between optimal dynamic and static pricing policies for two‐station tandem queues with finite buffers. Our paper extends the work of Wang et al () toward two directions: (a) Wang et al () does not offer any computational algorithms. We provide a tailored computational algorithm for each of three tandem queueing systems, from the (most general) system with a general intermediate buffer (ie, B 2 ∈ ℤ + ) and a general action set to the system with a general intermediate buffer and two prices, and then to the (most specific) system with zero intermediate buffer (ie, B 2 = 0) and two prices.…”