Consider the \"Bank of Quahog\" example in class. There are two clerks serving c

Question

Consider the "Bank of Quahog" example in class. There are two clerks serving customers in the bank (one clerk can only serve one customer at at a time). Suppose Peter enters an empty line but finds Brian and Stewie being served by the clerks. Peter will be served when one of the two clerks become available. Show that if each clerk serves a customer at an exponential rate lambda_i = i = 1, 2, then the probability that Peter is not last is the following: P{Peter is not last} = (lambda_1/lambda_1 + lambda_2)^2 + (lambda_2/lambda_1 + lambda_2)^2

Explanation / Answer

Solution: Suppose Clerk 1 serves Brian and Clerk 2 serves Stewe. Let S, P, B denote the respective service times for Stewe, Peter, and Brian. Then , there can be 2 conditions when this is possible:

1. Peter served by Clerk1 after he served Brian and Clerk2 continues serving Stewie

2. Peter served by Clerk2 after he served Stewie and Clerk1 continues serving Brian

Conditional Probability will be used here. Given the rates of service by Clerk1 and Clerk2 to be 1 and 2. The probability that serving time of Clerk1 is less than Clerk2 is given as:

P(B<S)= 1/( 1 + 2 )(Using continuous markov chain model)

Similiarly P(B>=S)= 2/( 1 + 2 ),

Now, using the conditional probability for Peter not being last, when B<S

P(Peter not last|B< S) = P(P < S)

(by the memoryless property, since Peter is not last, time taken to service Peter will be less than time take to service Stewie)

P (P<S)= 1/( 1 + 2 ).

Similarly, the law of total probability gives :

P(Peter not last) = P(Peter not last|B < S)P(B < S) + P(Smith not last|B S)P(B S)

= (1 /(1 + 2 ))· (1 /(1 + 2)) + (2 /(1 + 2)) · (2 /(1 + 2 ))

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