Question

Suppose you arrive at a post office having three clerks at a moment when both are busy but there is no one else waiting in line . You will enter service when either clerk becomes free. If service times for clerk i are exponential with rate λ1=1/hr , λ2=2/hr and λ3=3/hr , find E[T], where T is the amount of time that you spend in the post office

Answer 1

T is the amount of time that you spend in the post office

T = min(R1,R2,R3) + S

where

Let Ri = remaining service time of the customer with clerk i, i = 1, 2,3

S is your service time

Now

Ri follow Exponential distribution with i

min(R1,R2,R3) follow exponential distribution with rate = 1 + 2 + 3

E(min(R1,R2,R3) ) = 1/( 1 + 2 + 3) =1/( 1+2+3 ) = 1/6

S = exp(1) when R1 = min(R1 ,R2,R3)

=exp( 2) if R2 = min(R1 ,R2,R3)

= exp(3) if R3 = min(R1 ,R2,R3)

P(Rj = min (R1,R2,...,Rn))

= j/(1+ 2+ ...+n)

E(S) = E(S| R1 = min(R1,R2,R3)) P(R1 = min(R1,R2,R3)) + ....

= 1/1 * 1/(1+ 2+3) + ...

= 1/((1+ 2+ 3) + ...

E(S) = 3/( 1 + 2 + 3) = 3/6

hence

E(T) = E(min(R1,R2,R3)) + E(S)

= 1/6 + 3/6

=4/6

=2/3