Fluid Limits for Multiclass Many-Server Queues with General Reneging Distributions and Head-of-the-Line Scheduling

Abstract

We describe a fluid model with time-varying input that approximates a multiclass many-server queue with general reneging distribution and multiple customer classes (specifically, the multiclass G/GI/N+GI queue). The system dynamics depend on the policy, which is a rule for determining when to serve a given customer class. The class of admissible control policies are those that are head-of-the-line (HL) and nonanticipating. For a sequence of many-server queues operating under admissible HL control policies and satisfying some mild asymptotic conditions, we establish a tightness result for the sequence of fluid scaled queue state descriptors and associated processes and show that limit points of such sequences are fluid model solutions almost surely. The tightness result together with the characterization of distributional limit points as fluid model solutions almost surely provides a foundation for the analysis of particular HL control policies of interest. We leverage these results to analyze a set of admissible HL control policies that we introduce, called weighted random buffer selection (WRBS), and an associated WRBS fluid model that allows multiple classes to be partially served in the fluid limit (which is in contrast to previously analyzed static priority policies).

Asymptotically Optimal Idling in the GI/GI/N+GI Queue

“…Following the notation in Section 2.2 in [12], the state descriptor for the N -server queue is denoted by…”

“…As in [12], a state process for the N -server queue is a Y D valued, right continuous process Y N with finite left limits that satisfies a set of dynamic equations for the N -server queue consistent with HL service. See equations ( 5)-(26) in [12].…”

“…As in [12], a state process for the N -server queue is a Y D valued, right continuous process Y N with finite left limits that satisfies a set of dynamic equations for the N -server queue consistent with HL service. See equations ( 5)-(26) in [12]. In HL service, customers can only enter service at or after their arrival time and prior to their patience time expiring.…”

“…Once a server commences serving a customer, it works at rate one on the work associated with that customer until completely fulfilling that customer's service requirement, at which point the customer departs. The state process Y N is uniquely determined by an HL control policy which determines the number K N (t) of customers to enter service by time t, for each t > 0; see Section 2.4.1 and Definition 1 in [12] for further details. The initial condition Y N (0) for a state process Y N is assumed to be independent of the stochastic primitive inputs (see Section 2.1 in [12] for details).…”

“…The state process Y N is uniquely determined by an HL control policy which determines the number K N (t) of customers to enter service by time t, for each t > 0; see Section 2.4.1 and Definition 1 in [12] for further details. The initial condition Y N (0) for a state process Y N is assumed to be independent of the stochastic primitive inputs (see Section 2.1 in [12] for details). We denote the distribution of Y N (0) by ς N .…”