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Unformatted text preview: OPTIMALITY OF RANDOMIZED TRUNK RESERVATION FOR A PROBLEM WITH MULTIPLE CONSTRAINTS Xiaofei FanOrzechowski Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony Brook, NY 117943600 [email protected] (631) 6321487 Eugene A. Feinberg Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony Brook, NY 117943600 [email protected] (631) 6327189 submitted November 11, 2005 Abstract We study optimal admission of arriving customers to a Markovian finitecapacity queue, e.g. M/M/c/N queue, with several customer types. The system managers are paid for serving customers and penalized for rejecting them. The rewards and penalties depend on customer types. The penalties are modelled by a Kdimensional cost vector, K ≥ 1 . The goal is to maximize the average rewards per unit time subject to the K constraints on the average costs per unit time. Let K m denote min { K,m 1 } , where m is the number of customer types. For a feasible problem, we show the existence of a K mrandomized trunk reservation optimal policy, where the acceptance thresholds for different customer types are ordered ac cording to a linear combination of the service rewards and rejection costs. In addition, we prove that any K mrandomized stationary optimal policy has this structure. 1 Introduction and Problem Formulation In this paper, we consider a controlled finite capacity Markovian queue with m = 1 , 2 ,... types of customers arriving according to independent Poisson processes with the intensities λ i , i = 1 ,...,m, respectively. When a customer arrives, its type becomes known. When there are N customers in the system, the system is full and new arrivals are lost. If the system is not full, upon an arrival of a new customer, a decision of accepting or rejecting this customer is made. A positive reward r i is collected upon completion of serving an accepted type i customer. A nonnegative cost vector C i = ( C 1 ,i ,C 2 ,i ,...,C K,i ) τ incurs due to the rejection or lost of an arriving type i customer, where K is the number of constraints in this problem. The service time of a customer does not depend on the customer type. When there are n customers in the queue, the departure rate is μ n , n = 1 ,...,N. The numbers μ n , n = 1 ,...,N, satisfy the condition μ n 1 ≤ μ n , where μ = 0 and μ 1 > . In particular, for an M/M/c/N queue, for some μ > , μ i = ( iμ, if i = 1 ,...,c, cμ, if i = c + 1 ,...,N. 1 Unless otherwise specified, we do not assume that r 1 ≥ r 2 ≥ ··· ≥ r m . Our goal is to maximize the average rewards per unit time, subject to multiple constraints on average costs per unit time. This research is motivated by the following question: what is the structure of optimal policies for the problem when the blocking probabilities for some of the customer types do not exceed given numbers? The answer to this question is given in Corollary 4.3 below. Previously FanOrzechowskigiven numbers?...
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 Fall '11
 EugeneA.Feinberg
 Statistics, Linear Programming, Applied Mathematics, Optimization, LP, stationary optimal policy, Randomized Trunk Reservation, trunk reservation policy

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