3 Bank Robberies Chicago has been struck by a crime wave. Alarmed by the increas

Question

3 Bank Robberies Chicago 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: one teller's window is reserved for the exclusive use of bank robbers in order to conserve space, bank robberies may be committed only by a lone bandit if two or more robberies occur simultaneously, the robbers are served on a first-come, first-serve 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. The data you are given is: Robbers arrive at random 24/7 (during open hours), the average arrival rate is 2 robberies per hour Teller service is exponential with an average of 10 min (special robber withdrawal forms expedite service) You are asked to determine (a) the probability that there is no robber in the bank (large t probability (b) the probability that the robber's teller is busy. (c) the average number of robbers in the bank (d) the probability that a bank is hit at least 4 times by robberies during one hour (e) the probability that there is no robbery during a whole day. (f) the probability that the time until the fifth robbery exceeds two days (the original version of this problem is due to Shelly Weinberg of IBM) (8 points)

Explanation / Answer

This problem is a case of M/M/1 Queue system, where the robbers form customers and tellers provide the service.

Back-up Theory

An M/M/1 queue system is characterized by arrivals following Poisson pattern with average rate , service time following Exponential Distribution with average service rate of µ and single service channel.

Let n be the number of robbers in the bank at any given time.

Let (/µ) =

The steady-state probability of n robbers in the bank is given by Pn = n(1 - ) …………(1)

The steady-state probability of no customers in the system is given by P0 = (1 - ) ……(2)

Average number of robbers in the bank = E(n) = ()/(µ - )……………………………..(3)

Given = 2 per hour, µ = 6 [teller service on an average takes 10 minutes => average number of robbers serviced by the teller per hour = 60/10 = 6.] and hence = 2/6 = 1/3………. (4)

Part (a)

Probability there is no robber in the bank = P0 = (1 - ) [vide (3) of Back-up Theory]

= 1 – 1/3 [vide (4)]

= 2/3 ANSWER

Part (b)

Robbers’ teller will be busy so far as there is at least one robber in the bank. Hence, probability the robbers’ teller will be busy

= probability there is at least one robber in the bank

= 1 - probability there is no robber in the bank

= 1 - P0

= 1 - (1 - )

=

= 1/3 ANSWER

Part (c)

Average number of robbers in the bank

= E(n)

= ()/(µ - )

= 2/4

= ½ ANSWER

Part (d)

Probability bank is hit by at least 4 times by robberies during one hour

= P(X 4), where X = number of robberies during one hour which is given to follow Poisson Distribution with = 2 per hour.

So, the required probability, using Excel Function of Poisson Distribution,

= 0.1429 ANSWER

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