SEMI-MARKOV DECISION PROCESSES

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no more than M service requests remain in the system By doing so, we obtain a semi-Markov decision formulation with the state space I = {(i, t) | 0 i M, 0 t s}, and the action sets A(i, t) = {a | a = 0, , s}, {s}, 0 i M 1, 0 t s, i = M, 0 t s

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Here action a in state (i, t) means that the number of channels turned on is adjusted from t to a This semi-Markov decision formulation involves the following stipulation: if action a = s is taken in state (M, t), then the next decision epoch is de ned as the rst service completion epoch at which either M or M 1 service requests are left behind Also, if action a = s is taken in state (M, t), the onestep costs incurred until the next decision epoch are de ned as the sum of the switching cost K(t, s) and the holding and operating costs made during the time until the next decision epoch Denote by the random variable TM (s) the time until the next decision epoch when action a = s is taken in state (M, t) The random variable TM (s) is the sum of two components The rst component is the time until the next service completion or the next arrival, whichever occurs rst The second component is zero if a service completion occurs rst; otherwise, it is distributed as the time needed to reduce the number of service requests present from M + 1 to M The semi-Markov decision formulation with an embedded state space makes sense only when it is feasible to calculate the one-step expected transition times (M,t) (s) and the one-step expected costs c(M,t) (s) The calculation of these quantities is easy, since service completions occur according to a Poisson process with rate s as long as all of the s channels are occupied In other words, whenever M or more requests are in the system, we can equivalently imagine that a single superchannel is servicing requests one at a time at an exponential rate of s This analogy enables us to invoke the formulas (262) and (263) Taking n = 1 and replacing the mean by 1/(s ) in these formulas, we nd that the expected time needed to reduce the number of requests present from M + 1 to M, given that all channels are on, is 1 1/(s ) = 1 /(s ) s and the expected holding and operating costs incurred during the time needed to reduce the number of requests present from M + 1 to M, given that all channels are on, is hs 1 rs h(M +1) + rs h hM + + + = + s s s s (s ) s s (s )2 Here the term hM/(s ) represents the expected holding costs for the M service requests which are continuously present during the time needed to reduce

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OPTIMIZATION OF QUEUES

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the number in system from M + 1 to M If all of the s channels are busy, then the time until the next event (service completion or new arrival) is exponentially distributed with mean 1/( + s ) and the next event is generated by an arrival with probability /( + s ) Putting the pieces together, we nd (M,t) (s) = and c(M,t) (s) = K(t, s) + hM + rs + + s + s s + s h(M + 1) + rs h + s (s )2 + s 1 + + s + s 1 s = s ( + s )(s )

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Also, by the last argument above, p(M,t)(M 1,s) (s) = and p(M,t)(M,s) (s) =

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For the other states of the embedded state space I , the basic elements of the semi-Markov decision model are easily speci ed We have (i,t) (a) = and c(i,t) (a) = K(t, a) + hi + ra , + min(i, a) 0 i M 1, 0 a s 1 , + min(i, a) 0 i M 1, 0 a s,

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The one-step transition probabilities are left to the reader Next we formulate the value-iteration algorithm In the data transformation we take = 1/( + s ) Then the recurrence relation (723) becomes Vn ((i, t)) = min +

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