OPTIMIZATION OF QUEUES in .NET

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The semi-Markov model is a natural and powerful tool for the optimization of queues Many queueing problems in telecommunication ask for the computation of an optimal control rule for a given performance measure If the control rule is determined by one or two parameters, one might rst use Markov chain analysis to calculate the performance measure for given values of the control parameters and next use a standard optimization procedure to nd the optimal values of the control parameters However, this is not always the most effective approach Below we give an example of a controlled queueing system for which the semi-Markov decision approach is not only more elegant, but is also more effective than a direct search procedure In this application the number of states is unbounded However, by exploiting the structure of the problem, we are able to cast the problem into
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OPTIMIZATION OF QUEUES
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a Markov decision model with a nite state space Using a simple but generally useful embedding idea, we avoid brute-force truncation of the in nite set of states Example 741 Optimal control of a stochastic service system A stochastic service system has s identical channels available for providing service, where the number of channels in operation can be controlled by turning channels on or off For example, the service channels could be checkouts in a supermarket or production machines in a factory Requests for service are sent to the service facility according to a Poisson process with rate Each arriving request for service is allowed to enter the system and waits in line until an operating channel is provided The service time of each request is exponentially distributed with mean 1/ It is assumed that the average arrival rate is less than the maximum service rate s A channel that is turned on can handle only one request at a time At any time, channels can be turned on or off depending on the number of service requests in the system A non-negative switching cost K(a, b) is incurred when adjusting the number of channels turned on from a to b For each channel turned on there is an operating cost at a rate of r > 0 per unit of time Also, for each request a holding cost of h > 0 is incurred for each unit of time the message is in the system until its service is completed The objective is to nd a rule for controlling the number of channels turned on such that the long-run average cost per unit of time is minimal Since the Poisson process and the exponential distribution are memoryless, the state of the system at any time is described by the pair (i, t), where i = the number of service requests present, t = the number of channels being turned on The decision epochs are the epochs at which a new request for service arrives or the service of a request is completed In this example the number of possible states is unbounded since the state variable i has the possible values 0, 1, A brute-force approach would result in a semi-Markov decision formulation in which the state variable i is bounded by a suf ciently large chosen integer U such that the probability of having more than U requests in the system is negligible under any reasonable control rule This approach would lead to a very large state space when the arrival rate is close to the maximum service rate s A more ef cient Markov decision formulation is obtained by restricting the class of control rules rather than truncating the state space It is intuitively obvious that under each reasonable control rule all of the s channels will be turned on when the number of requests in the system is suf ciently large In other words, choosing a suf ciently large integer M with M s, it is from a practical point of view no restriction to assume that in the states (i, t) with i M the only feasible action is to turn on all of the s channels However, this implies that we can restrict the control of the system only to those arrival epochs and service completion epochs at which
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