A Fair Policy for theG/GI/NQueue with Multiple Server Pools

Abstract

We consider the G/GI/N queue with multiple server pools, each possessing a pool-specific service time distribution. The class of nonidling routing policies that we consider are referred to as u-greedy policies. These policies route incoming customers to the server pool with the longest weighted cumulative idle time to equitably spread incoming work amongst the server pools in the system. Our first set of results demonstrates that asymptotically in the Halfin-Whitt regime and under any u-greedy policy, the diffusion scaled cumulative idle time processes of each of the server pools are held in fixed proportion to one another. We next provide a heavy traffic limit theorem for the process keeping track of the total number of customers in the system. Our limit may be characterized as the solution to a stochastic convolution equation that is driven by a Gaussian process. To prove our main results, we introduce a new methodology for studying the G/GI/N queue in the Halfin-Whitt regime that has as its starting point a simple conservation of flow identity. Introduction. In the literature recently (Armony and Ward [3], Atar [4], Atar et al. [5], Gurvich andWhitt [18]), many papers have been published on the topic of providing fairness in many-server queueing systems with multiple server pools, with the concept of fairness being directed toward the customers as well as toward the servers. The majority of these papers focus on many-server queueing systems in the Halfin-Whitt [19] regime with exponentially distributed service times, possibly with service rates varying across the different server pools. The results of these papers make it overwhelmingly clear that under the assumption of exponentially distributed service times in each server pool, it is possible to achieve a high level of performance in a many-server queue while at the same time equitably sharing the incoming work across the different servers in the system.One of the primary motivating applications for the study of many-sever queues is the design and control of large-scale telephone call centers. However, it has been well documented in Brown et al.[8] that service times in telephone call centers are not exponentially distributed but rather appear to be lognormally distributed. This then brings us to the following important set of questions. Which policies already demonstrated to perform well for many-server queues under the assumption of exponentially distributed service times also perform well for general service time distributions? In other words, how robust are these policies with regard to changes in the service time distribution? Moreover, given that a policy performs well in the case of general service time distributions, how may we begin to evaluate its detailed performance measures?In the present paper, the above set of questions is addressed by considering a many-server queue with a single class of customers and multiple server pools. This type of system is commonly referred to in the literature as the inverted-V model. We assume that service times wi...

Routing and staffing when servers are strategic

“…The issue from an operational standpoint is that there is strong indication in the HRM literature that the perception of fairness affects employee performance (see [15], [14]). This has motivated the analysis of "fair" routing policies that, for example, equalize the cumulative server idleness ( [7,36]), and the desire to find an optimal "fair" routing policy ( [5,40]). Another approach is to formulate a model in which the servers choose their service rate in order to balance their desire for idle time (which is obtained by working faster) and the exertion required to serve faster.…”

Routing and Staffing When Servers Are Strategic

“…The issue from an operational standpoint is that there is strong indication in the human resource management literature that the perception of fairness affects employee performance [15,14]. This has motivated the analysis of "fair" routing policies that, for example, equalize the cumulative server idleness [7,38], and the desire to find an optimal "fair" routing policy [5,42]. Another approach is to formulate a model in which the servers choose their service rate in order to balance their desire for idle time (which is obtained by working faster) and the exertion required to serve faster.…”

A Unified Approach to Diffusion Analysis of Queues with General Patience-Time Distributions

“…By Proposition 2.1 in Reed (2009), similar to Proposition 4.4 of Reed and Shaki (2015), and in view of (3.14), we havẽ…”

“…The idea of (3.14), which follows Equation ( 33) in Reed and Shaki (2015), is to center the service completion process using the renewal function M (•). Then (3.12) becomes…”