Collaboration versus Specialization in Service Systems with Impatient Customers

TL;DR

The paper investigates tandem queues with flexible, synergistic servers and impatient customers, focusing on dynamic server assignment to maximize long-run throughput. It proves that for generalist servers the expedite policy (all servers collaborate on the same job) is optimal and provides a complete threshold-based policy characterization for a 2-station, 2-server Markovian system. It further analyzes how the optimal policy varies with synergy $\gamma$ and abandonment rate $\theta$, and extends the results to task-dependent synergy, offering a practical policy and numerical validation. The findings emphasize that higher synergy reduces the need for buffering, while abandonments incentivize more aggressive server collaboration, with the expedite policy often prevailing in many realistic settings.

Abstract

We study tandem queueing systems in which servers work more efficiently in teams than on their own and customers are impatient in that they may leave the system while waiting for service. Our goal is to determine the server assignment policy that maximizes the long-run average throughput. We show that when each server is equally skilled at all tasks, the optimal policy has all the servers working together at all times. We also provide a complete characterization of the optimal policy for Markovian systems with two stations and two servers when each server's efficiency may be task dependent. We show that the throughput is maximized under the policy which assigns one server to each station (based on their relative skill at that station) unless station 2 has no work (in which case both servers work at station 1) or the number of customers in the buffer reaches a threshold whose value we characterize (in which case both servers work at station 2). We study how the optimal policy varies with the level of server synergy (including no synergy) and also compare the optimal policy for systems with different customer abandonment rates (including no abandonments). Finally, we investigate the case where the synergy among collaborating servers can be task-dependent and provide numerical results.

Figures (4)

• Figure 1: Behavior of the long-run average throughput $g_n$ as a function of $n$$(B=5)$.
• Figure 2: Behavior of $N$ as a function of $\gamma \in [1,2]$ and $\theta \in [0,2]$.
• Figure 3: Domains of the optimal buffer size $\bar{N}$ as functions of $\gamma_1$ and $\gamma_2$.
• Figure 4: Ranges of Parameters.

Theorems & Definitions (14)

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