Technical Note: What's in a Constraint? On the Ambiguity of Standard Delay Targets

• 成果类型: Article; Early Access
• 署名作者: Soh, Seung Bum; Gurvich, Itai
• 署名单位: Yonsei University; Northwestern University
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2022.0572
• 发表日期: 2025
• 关键词: call centers; Asymptotic Optimality; Waiting time; QUEUE; policies; Servers

摘要

Staffing problems are often formulated as satisfization problems, in which the cost of servers is minimized subject to quality of service constraints. These constraints indirectly capture customers' disutility from waiting or, at least, its structure. For the problem of staffing a single-class M/M/N queue with an average speed of answer (ASA) constraint, any work-conserving policy is optimal; the problem's formulation is, in that sense, ambiguous. One optimal solution is consistent with convex delay disutilities: the first in, first out (FIFO) policy, but another optimal solution uses last in, first out, which is consistent with concave costs. FIFO and, in turn, a representation of convex delay disutility arises as the unique optimal solution if one insists on variance minimality. Imposing FIFO is another way to guarantee that the solution is consistent with convex disutilities. We formalize this understanding of ambiguity and extend it to a multiclass environment. It is shown that ambiguity persists even if one insists on FIFO within each of the customer classes. An arrival rate-weighted index of dispersion is introduced as the generalization of variance that disambiguates the formulation. Choosing the one that minimizes it makes the formulation unambiguous and representative of convex disutilities. In the same vein, we show that fixed queue ratio policies are the generalization of FIFO. Imposing the use of a policy from this family guarantees a solution that is consistent with convex delay disutilities. Characterizing ambiguity requires mapping the full space of solutions-instead of providing a single solution-to the multiclass satisfization problem. We perform this characterization of the solution space for a variety of constraints: convex, linear (ASA), and concave.