Open Problem—Regarding Static Priority Scheduling for Many-Server Queues with Reneging

We study the many-server queue shown in Figure 1(a). Customers from class j ∈ {1, 2} arrive according to renewal processes having rate λ_j > 0 and request processing. There are N ∈ {1, 2,…} servers, each working at rate one. All servers are fully flexible; that is, every server can serve customers from both classes. Each customer has a randomly distributed patience time that may depend on the class and reneges (abandons the system without receiving service) if service does not begin before the patience time expires. The system incurs the penalty p₁ > 0 when a class 1 customer reneges and the penalty p₂ > 0 when a class 2 customer reneges. Upon arrival, each class j customer independently samples from the distribution determined by cumulative distribution function (cdf) $G_j^r$ having mean 1/θ_j < ∞ to find its patience time and from the distribution determined by cdf $G_j^s$ having mean 1/μ_j < ∞ to determine its service time (that is, the required processing time).