Los Angeles has been struck by a crime wave. Alarmed by the increasing number of bank robberies and concerned about their effect on bank customers, the Banking Upper Management Society (BUMS) adopts the following policies at each bank:
(a) A teller's window is reserved for the exclusive use of bank robbers.
(b) In order to conserve space, bank robberies may be committed only by a lone bandit.
(c) If two or more robberies occur simultaneously, the robbers are served on a first-come, first-served basis.
You are engaged as a consultant by the Bank Robbers Federation (BARF). Your job is to determine if the proposed arrangement with the BUMS is adequate. [Please keep in mind the type of overshoes you are likely to be wearing if you don't get this right.] The data you are given is:
(i) Robbers arrive at random between the hours of 9:00 a.m. and 3:00 p.m.; the average arrival rate is five robbers per hour.
(ii) Teller service time is exponential with an average value of 2 minutes (for the robber's teller). (Special robber withdrawal forms expedite service.)
(iii) The $\mathrm{M} / \mathrm{M} / 1$ model seems to apply.
You are asked to determine
(1) the average time a robber must queue for service (a robbery).
(2) the average time required for a robbery (queueing time plus service time).
(3) the probability the robber's teller is busy.
(4) the average number of robbers in the bank.
(5) the probability of finding three or more robbers in the bank at the same time.
(6) the probability a robber spends more than 15 minutes in the bank.
(7) the 95 th percentile of robbery time.
(The original version of this problem is due to Shelly Weinberg of IBM.)