FanFeinberg1 - Applied Probability Trust(9 January 2006...

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Applied Probability Trust (9 January 2006) OPTIMALITY OF RANDOMIZED TRUNK RESERVATION FOR A PROBLEM WITH A SINGLE CONSTRAINT XIAOFEI FAN-ORZECHOWSKI, * State University of New York at Stony Brook EUGENE A. FEINBERG, ** State University of New York at Stony Brook Abstract We study an optimal admission of arriving customers to a Markovian finite- capacity 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 goal is to maximize the average rewards per unit time subject to the constraint on the average penalties per unit time. We provide a solution to this problem based on Lagrangian optimization. For a feasible problem, we show the existence of a randomized trunk reservation optimal policy with the acceptance thresholds for different customer types ordered according to a linear combination of the service rewards and rejection costs. In addition, we prove that any 1-randomized stationary optimal policy has this structure. In particular, we establish the structure of an optimal policy that maximizes the average rewards per unit time subject to the constraint on the blocking probability for one of the customer types or for a group of customer types pooled together. Keywords: queue; admission control; semi-Markov decision process; trunk reservation; Lagrangian optimization 2000 Mathematics Subject Classification: Primary 60K25 Secondary 60K15; 60K30 1. Introduction This paper describes the structure of optimal admission policies to finite capacity queues, including M/M/c/N queues, with a fixed number of customer types. At the arrival epoch a customer can be either rejected or accepted. The latter is possible only if the system is not full. Each customer type i = 1 , 2 , . . . , m, where m is the number of customer types, is characterized by three parameters: a Poisson arrival rate λ i , a reward r i that a customer pays for the service, and a penalty c i paid to a rejected customer. The service times do not depend on the customer types. The goal is to maximize the average rewards per unit time subject to the constraint that the average penalty per unit time does not exceed a certain number. Such problems arise, for example, when the goal is to maximize the average rewards per unit time subject to the quality of service (QoS) constraint. * Postal address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY, 11794-3600. Email: [email protected] ** Postal address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook ,NY, 11794-3600. Email: [email protected] 1
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2 X. FAN-ORZECHOWSKI and E. A. FEINBERG A randomized trunk reservation policy φ is defined by m numbers M φ i , 0 M φ i N - 1 , i = 1 , . . . , m. Among these numbers M φ 1 , . . . , M φ m , at most one number is non- integer and at least one number equals N - 1 . For a number M we denote by b M c the integer part of M . If the system is controlled by the policy φ , a type
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