5.10 Hypercube approximation procedure for the zero-line-capacity case Suppose that we wish to derive an approximate procedure for finding the performance measures of the hypercube model, analogous to that of Section 5.5, but assuming zero line capacity. To do this, we must develop a new Q factor and new workload normalization conditions.

1. Verify that the appropriate steady-state probabilities for the corresponding M / M / N zero line capacity queue are
   
   
   
   where /N < + (Assume that = 1)

2. Confirm that the average utilization factor is
   
   
   
   

3. Now we would like to develop a correction factor Q'(N, , j) that, when multiplied by ^j(1 - ), gives the exact probability P{B₁B₂. . .B_jF_j+1} for the M / M / N zero line capacity system. Following reasoning analogous to Problem 5.9, verify that
   
   
   
   and where Q^*(N, , j) is equal to Q(N, , j)as computed for the M / M / N infinite line capacity case, but with P{S_o} replaced by P'{S_o}

4. Conclude that an appropriate workload approximation procedure for the zero-line-capacity case would utilize (5.52), (5.53), and the algorithm of Figure 5.19, with Q( ) replaced by Q'( ) and with the following other modifications: