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The M/M/s queueing process can be seen as a Markov decision process with a single decision in each state The decision is to leave the system alone In this Markov decision formulation it is convenient to consider the state of the system both at the arrival epochs and the service completion epochs In the M/M/s queue the situation of i customers present just after a service completion is probabilistically the same as the situation of i customers present just after an arrival In accordance with (631), we de ne the relative cost function w(i) by w(i) = where T (i) = the expected time until the rst return to an empty system starting with i customers present, K(i) = the total expected cost incurred until the rst return to an empty system starting with i customers present Then, by the economic interpretation of the relative values given in Section 63, we have for any i = 0, 1, that w(i + 1) w(i) = the difference in total expected costs over an in nitely long period of time by starting in state i + 1 rather than in state i The desired function Dk (i) for queue k follows by taking Dk (i) = wk (i + 1) wk (i) with = pk , s = sk and = k K(i) gT (i), i = 1, 2, , 0, i = 0, (752)
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The basic functions K(i) and T (i) are easy to compute By conditioning, Ti = Ki = 1 i + Ti+1 + Ti 1 , + i + i + i i i + Ki+1 + Ki 1 , + i + i + i 1 i s 1 i s (753) (754)
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where T0 = K0 = 0 Further, we have Ti = Ki = i s + Ts , s 1 s i > s, i > s
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To see the latter relations, note that the time to reach an empty system from state i > s is the sum of the time to reach state s and the time to reach an empty system from state s By the memoryless property of the exponential distribution, the multiserver M/M/s queue operates as a single-server M/M/1 queue with service rate s when s or more customers are present Next, by applying the formulas (262) and (263), we nd the formulas for Ti and Ki when i > s Substituting the expressions
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for Ts+1 and Ks+1 into (753) and (754) with i = s, we get two systems of linear equations for Ti , 1 i s and Ki , 1 i s Once these systems of linear equations have been solved, we can next compute Ti and Ki for any desired i > s Summarizing, the heuristic algorithm proceeds as follows Heuristic algorithm
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(0) Step 1 Compute the best values pk , k = 1, , n, of the Bernoulli-splitting probabilities by minimizing the expression (751) subject to n pk = 1 and k=1 0 pk < sk k for k = 1, , n Step 2 For each queue k = 1, , n, solve the system of linear equations (753) (0) and (754) with = pk , s = sk and = k Next compute for each queue k (0) the function wk (i) from (752) with = pk , s = sk and = k Step 3 For each state x = (i1 , , in ), determine an index k0 achieving the minimum in min {wk (ik + 1) wk (ik )} 1 k n
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The separable rule assigns a new arrival in state x = (i1 , , in ) to queue k0 Numerical results Let us consider the numerical data s1 = 10, s2 = 1, 1 = 1 and 2 = 9
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The traf c load , which is de ned by = /(s1 1 + s2 2 ), is varied as = 02, 05, 07, 08 and 09 In addition to the theoretically minimal average sojourn time, Table 751 gives the average sojourn time per customer for the Bernoulli-splitting rule (B-split) and for the heuristic separable rule The table also gives the average sojourn time per customer under the shortest expected delay (SED) rule Under this rule an arriving customer is assigned to the queue in which its expected individual delay is smallest (if there is a tie, the customer is sent to queue 1) The results in the table show that this intuitively appealing control policy performs unsatisfactorily for the case of heterogeneous services However, the heuristic separable rule shows an excellent performance for all values of
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Table 751 02 05 07 08 09 SED 0192 0647 0883 0982 1235 The average sojourn times B-split 0192 0579 0737 0897 1404 Separable 0191 0453 0578 0674 0941 Optimal 0191 0436 0575 0671 0931
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