Consider a local shop operating on a 12-hour daily shift that serves orders arriving according to a Poisson process with rate = 5 per day. The shop processes the incoming orders according to a FCFS protocol, one order at a time, and each order is filled by manufacturing and delivering a certain part. The manufacturing of a part takes two hours, but a part can be damaged while in processing, in which case, a new part must be started for the satisfaction of the corresponding order.

Assuming that such a catastrophic failure can occur uniformly over the 2- hour interval that is required for the complete processing of the part, do the following:

i. Compute the maximum failing probability p for the processing of any single part that will lead to a stable operation of the considered shop.

ii. Perform a mean-value analysis of the shop operation assuming that the failing probability is p = 0.5 p. In particular, provide the utilization, the throughput, the expected lead time for an order, and the average number of standing orders at any time point.

Remark: Remember that a compound random variable (r.v.) is a r.v. Y = NX i=1 Xi where Xi are i.i.d. r.v.s and N is another random variable independent of Xi. For r.v. Y we have: E[Y ] = E[N ] E[Xi] and V ar[Y ] = E[N ] V ar[Xi] + V ar[N ] E2[Xi] You will need these results for the modeling of the effective processing time of the orders that are filled by the shop.