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The policy-iteration algorithm has the remarkable feature that it achieves the largest improvements in costs in the rst few iterations These ndings underlie a heuristic approach for Markov decision problems with a multidimensional state space In such decision problems it is usually not feasible to solve the value-determination equations However, a policy-improvement step offers in general no computational dif culties This suggests a heuristic approach that determines rst a good estimate for the relative values and next applies a single policy-improvement step By the nature of the policy-iteration algorithm one might expect to obtain a good decision rule by the heuristic approach How to compute the relative values to be used in the policy-improvement step typically depends on the speci c application The heuristic approach is illustrated in the next example Example 751 Dynamic routing of customers to parallel queues An important queueing model arising in various practical situations is one in which arriving customers (messages or jobs) have to be assigned to one of several different groups of servers Problems of this type occur in telecommunication networks and exible manufacturing The queueing system consists of n multi-server groups working in parallel, where each group has its own queue There are sk servers in group k (k = 1, , n) Customers arrive according to a Poisson process with rate Upon arrival each customer has to be assigned to one of the n server groups The assignment is irrevocable The customer waits in the assigned queue until a server becomes available Each server can handle only one customer at a time The problem is to nd an assignment rule that (nearly) minimizes the average sojourn time per customer This problem will be analysed under the assumption that the service times of the customers are independent and exponentially distributed The mean service time of a customer assigned to queue k is 1/ k (k = 1, , n) It is assumed that < n sk k In what follows we consider the minimization k=1 of the overall average number of customers in the system In view of Little s formula, the minimization of the average sojourn time per customer is equivalent to the minimization of the average number of customers in the system Bernoulli-splitting rule An intuitively appealing control rule is the shortest-queue rule Under this rule each arriving customer is assigned to the shortest queue Except for the special case of s1 = = sn and 1 = = n , this rule is in general not optimal In particular, the shortest-queue rule may perform quite unsatisfactorily in the situation

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of a few fast servers and many slow servers Another simple rule is the Bernoullisplitting rule Under this rule each arrival is assigned with a given probability pk to queue k (k = 1, , n) irrespective of the queue lengths This assignment rule produces independent Poisson streams at the various queues, where queue k receives a Poisson stream at rate pk The probabilities pk must satisfy k pk = 1 and pk < sk k for k = 1, , n This condition guarantees that no in nitely long queues can build up Under the Bernoulli-splitting rule it is easy to give an explicit expression for the overall average number of customers in the system The separate queues act as independent queues of the M/M/s type This basic queueing model is discussed in 5 In the M/M/s queue with arrival rate and s exponential servers each with service rate , the long-run average number of customers in the system equals (s )s L(s, , ) = s!(1 )2

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