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The closed two-node network with this blocking protocol behaves identically to the nite-source model Thus the nite-source model has the insensitivity property This result provides a simple but heuristic argument that the controlled loss model with Poisson input also has the insensitivity property In general, insensitivity holds for a wide class of loss networks; see Kelly (1991) and Ross (1995) Let us now assume exponentially distributed transmission times for the loss model controlled by an L-policy De ne Xj (t) = the number of channels occupied by type j messages at time t for j = 1, 2 The stochastic process {(X1 (t), X2 (t))} is a continuous-time Markov chain with state space I = {(i1 , i2 ) | 0 i1 + i2 c, i1 0, 0 i2 L} Its transition rate diagram is given in Figure 541 By equating the rate out of state (i1 , i2 ) to the rate into state (i1 , i2 ), we obtain the equilibrium equations for the
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Figure 541 The transition rate diagram for the L-rule
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state probabilities p(i1 , i2 ) For the states (i1 , i2 ) with i1 + i2 < c and i2 < L, (i1 1 + i2 2 + 1 + 2 )p(i1 , i2 ) = 1 p(i1 1, i2 ) + 2 p(i1 , i2 1) + (i1 + 1) 1 p(i1 + 1, i2 ) + (i2 + 1) 2 p(i1 , i2 + 1) For the states (i1 , i2 ) with i1 + i2 < c and i2 = L, (i1 1 + i2 2 + 1 )p(i1 , i2 ) = 1 p(i1 1, i2 ) + 2 p(i1 , i2 1) + (i1 + 1) 1 p(i1 + 1, i2 ) For the states (i1 , i2 ) with i1 + i2 = c and i2 L, (i1 1 + i2 2 )p(i1 , i2 ) = 1 p(i1 1, i2 ) + 2 p(i1 , i2 1) The state probabilities p(i1 , i2 ) exhibit the so-called product form p(i1 , i2 ) = C ( 1 / 1 )i1 ( 2 / 2 )i2 , i1 ! i2 ! i1 , i2 I
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for some constant C > 0 The reader may verify this result by direct substitution into the equilibrium equations Since service completions occur in state (i1 , i2 ) at a rate of i1 1 + i2 2 , the average throughput is given by T (L) = (i1 1 + i2 2 )p(i1 , i2 )
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Denote by j (L) the long-run fraction of type j messages that are lost Using the PASTA property, it follows that
c L 1 1 (L) = (i1 ,i2 ): i1 +i2 =c
p(i1 , i2 )
2 (L) = i1 =0
p(i1 , L) +
(i1 ,i2 ): i1 +i2 =c
p(i1 , i2 )
Since the sum of the average number of messages lost per time unit and the average number of messages transmitted per time unit equals the arrival rate 1 + 2 , we have the identity 1 1 (L) + 2 2 (L) + T (L) = 1 + 2 This relation is useful as an accuracy check for the calculated values of the p(i1 , i2 ) As an illustration, we consider the following numerical data: c = 10, 1 = 10, 2 = 7, 1 = 10, 2 = 1 Table 541 gives the values of T (L), 1 (L) and 2 (L) for L = 7, 8 and 9 The L-policy with L = 8 maximizes the long-run average throughput among the class of L-policies The above analysis restricted itself to the easily implementable L-policies, but other control rules are conceivable The question of how to compute the overall optimal control rule among the class of all conceivable control rules will be addressed in the s 6 and 7, which deal with Markov decision processes The best L-policy is in general not optimal among the class of all possible control rules However, numerical investigations indicate that using the best L-policy rather than the overall optimal policy often leads to only a small deviation from the theoretically maximal average throughput For example, for the above numerical data the average throughput of 15209 for the best L-policy is only 016% below the theoretically optimal value of 15233 This optimal value is achieved by the following control rule Each arriving message of type 1 is accepted as long as not all channels are occupied A message of type 2 nding i messages of type 1 present upon arrival is accepted only when less than Li other messages of the same type 2 are present and not all of the channels are occupied The optimal values of the Li are L0 = L1 = 8, L2 = L3 = 7, L4 = 6, L5 = 5, L6 = 4, L7 = 3, L8 = 2 and L9 = 1 The insensitivity property is no longer exactly true for the Li -policy, but numerical investigations indicate that the dependency on the distributional form of the transmission times is quite weak The above Li -policy 2 was simulated for lognormally distributed transmission times Denoting by ci the squared coef cient of variation of the transmission time for type i messages, we 2 2 varied (c1 , c2 ) as (1, 1), (2, 05) and (05, 2) For these three examples the average