7.4 Modeling the Workstation Following a Batch Server
CTq (2) = =
CB2 [Ta (2)] +C 2[Ts (B)] 2
u2 1 − u2
217
E[Ts (B)]
CB2 [Ta (2)] +C 2[N] +C 2[Ts (I)]/E[N] 2
u2 1 − u2
(7.22) E[N]E[Ts(I)] ,
where the utilization factor is computed by u2 = λ2 (I)E[Ts (I)]. • Suggestion: Do Problem 7.13. 7.4.2.2 Average Service Delay Times for Random Sized Batches Once a batch has worked its way through the batch queue and finally has command of the server, the server will be busy for the specific number of service times equal to the number of items in the batch. The delay time associated with individual items within the batch varies since processed items leave the workstation immediately upon completion of their turn in the server. Thus, an average delay is computed by taking into account the delay associated with each position, with respect to the order that items are served, within the batch. This average delay has two components, the service time of the individual and the average delay waiting for other items positioned ahead of that individual unit in the batch. We follow the same logic here that was used in Sect. 7.1.3. The random variable D represents the total delay experienced by all jobs within a batch; thus from Eq. (7.5), it follows that E[D|N = n] =
n(n − 1) E[Ts (I)] , 2
where the random variable N is the size of the batch. Since E[ E[D|N] ] = E[D] (see Property 1.8), the service time delay, st, is obtained as follows st =
E[N(N − 1)] E[D] + E[Ts (I)] = E[Ts (I)] + E[Ts (I)] E[N] 2E[N]
(7.23)
E[N(N + 1)] E[N 2 ] − E[N]2 + E[N]2 + E[N] E[Ts (I)] = E[Ts (I)] 2E[N] 2E[N] E[N] + 1 V [N] 1 + E[N] + E[N]C 2[N] = + E[Ts (I)] = E[Ts (I)] . 2 2E[N] 2
=
Notice that for deterministic batches, Eq. (7.23) is identical to Eq. (7.6).