Optimal buffer size for a stochastic processing network in heavy traffic

Abstract

We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The controller can choose the drift coefficient which represents the service rate and the buffer size b > 0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A "control cost" related to the dynamically controlled service rate, a "congestion cost" which depends on the queue length and a "rejection penalty" for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size b * > 0. When the buffer size b > 0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145-1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145-1160, 2005).To obtain a solution to the corresponding Hamilton-JacobiBellman (HJB) equation, we analyze a family of ordinary A.P. Ghoshdifferential equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b * > 0.

Optimal Admission Control for Many-Server Systems with QED-Driven Revenues

“…Threshold control policies are found to be optimal in a variety of contexts in such as De Waal (1990), Chen and Frank (2001), Bekker and Borst (2006), and many (implicit) characterizations of optimal threshold values have been obtained in Naor (1969), Yildirim and Hasenbein (2010), and Borgs et al (2014). For (single-server) queues in a conventional heavy-traffic regime, optimality of threshold control policies has been established by studying limiting diffusion control problems in Ghosh and Weerasinghe (2007), Ward and Kumar (2008), and Ghosh and Weerasinghe (2010). The analysis of control problems in the QED regime has mostly focused on routing and scheduling, see Atar et al (2004), Atar (2005a, b), Atar et al (2006), and Gurvich and Whitt (2009 Stochastic Systems, 2017, vol.…”

Admission control for a multi-server queue with abandonment

“…There have been several recent papers that study admission control problems in the conventional heavy traffic regime (see, for example, [13] when there is no customer abandonment, and [14,22] when there is customer abandonment). However, the only other work we find that considers an admission control problem in the Halfin-Whitt (see [15]) limit regime, extended by Garnett et al [11] to include customer abandonment, is that of Weerasinghe and Mandelbaum [23].…”

“…13) whereX(∞) has the steady-state density associated with the processX in(4.1) under controlÛ 0 . The expressions for E[X(∞)|X(∞) ≤ 0] and E[X(∞)|X(∞) > 0] are given in (18.29) of Browne and Whitt…”

Controlled Stochastic Dynamical Systems

“…The problem addressed in [7] is to find an optimal control policy (X * x , u * , U * ), for the control problem (2.4) under reasonable assumptions and to characterize the value v 0 in (2.4). In addition to the previous assumptions, we also assume that the control set A is a subset of [0, ∞) and it contains the interval [0, θ 0 ] where θ 0 satisfies C (θ 0 ) = 1.…”

“…This work is motivated by our previous work in [7] and also by the recent articles of [18] and [20]. The model here is similar to that of (2.1) but in addition to that impatient customers are allowed to leave the queue at a rate of γ > 0.…”