Dynamic service rate control for a single-server queue with Markov-modulated arrivals

Abstract

Abstract:We consider the problem of service rate control of a single-server queueing system with a finite-state Markov-modulated Poisson arrival process. We show that the optimal service rate is nondecreasing in the number of customers in the system; higher congestion levels warrant higher service rates. On the contrary, however, we show that the optimal service rate is not necessarily monotone in the current arrival rate. If the modulating process satisfies a stochastic monotonicity property, the monotonicity is recovered. We examine several heuristics and show where heuristics are reasonable substitutes for the optimal control. None of the heuristics perform well in all the regimes and the fluctuation rate of the modulating process plays an important role in deciding the right heuristic. Second, we discuss when the Markov-modulated Poisson process with service rate control can act as a heuristic itself to approximate the control of a system with a periodic nonhomogeneous Poisson arrival process. Not only is the current model of interest in the control of Internet or mobile networks with bursty traffic, but it is also useful in providing a tractable alternative for the control of service centers with nonstationary arrival rates.

Stationary distributions and convergence for M/M/1 queues in interactive random environment

“…The paper [41] deals with similar questions by comparing this queue with an appropriate M/M/1 queue. Optimization of service rate for the case when arrival rate is a Markov process is studied in [26]. See also, a birthdeath process in random environment [13] and a Markov chain in Markov environment, studied in [12,16,33].…”

Combined variance reduction techniques in fully sequential selection procedures

“…The customer arrival rate is constant but different for each system . We want to allow the server to become faster when there are more customers waiting in queue, which is referred as the dynamic service rate control policy in the literature (eg, Kumar, Lewis, & Topaloglu, ). The service rate for customer from system is specified as , where is random and represents the number of customers waiting in line when the th customer just enters the server.…”

Value of capacity flexibility in manufacturing systems with seasonal demands

“…The system responds to the fluctuating demand through dynamic pricing and inventory replenishment. Kumar et al (2013) analyze service facilities with exponentially distributed production times and seasonal demands. Using dynamic programming, they identify the optimal policy as a state-dependent production rate.…”

“…We consider a policy π such that μ n,s = μ max for n < N 0 and μ n,s = 0 for n ≥ N 0 for all s ∈ S, where N 0 is a positive integer. Then using arguments similar to Proposition 2.2 of Kumar et al (2013), it can be shown that under the condition described in Equation (1), the Markov chain induced due to this policy is irreducible, ergodic in the state space (−∞, N 0 ] × S, and has finite long-run expected average cost. Note that we have validated Assumption 7 under finite upper bound of the net inventory level.…”