Question

solve using excel A-C

8. The parts department of a large automobile dealership has a counter used exclusively for mechanics' requests for parts. The time between requests can be modeled by a negative exponential distribution that has a mean of five minutes. A clerk can handle requests at a rate of 15 per hour, and this can be modeled by a Poisson distribution that has a mean of 15. Suppose there are two clerks at the counter. a. On average, how many mechanics would be at the counter, including those being served? b.What is the probability that a mechanic would have to wait for service? c. If a mechanic has to wait, how long would the average wait be? d. What percentage of time are the clers idle? e. If clerks represent a cost of $20 per hour and mechanics a cost of $30 per hour, what number of clerks would be optimal in terms of minimizing total cost?

Answer 1

15 per hour 15 per hour Average arrival rate (A) Average service rate (u) No. of servers (s) Answers (a) (b) (c) (d) 1.333 2 0.333 1.333 minutes 1 33.33% 1 2 n! n-0 S.l S.u-2 Po= 0.333 0.333 Lg L 1.333 Pr 0.333 W 0.022 hours 1.333 minutes

Computations

A B D F G H 1 per hour per hour Average arrival rate (A) Average service rate (u) No. of servers (s) 15 Answers 2 15 (a) (b) (c) |(d) C11 3 -C 12 2 C2/C3 -C14 minutes 5 =C9 6 1 = 1*C5^0 (1/1)*C5^1 n! 12 s! u} s.i-1-1/2)*C5^C4*(C4*C3) / (C4*C3 - C2) S.l 8 Po=1/(C7+C8) 9 Lg C2*C3*(C5^C4)*C9/(FACT(C4-1)*(C4*C3-C2)2) L C10+C5 P C10*(C4/C5 - 1) 10 11 12 Wa = =C10/C2 hours 13 -C13*60 minutes 14 - 7

(e)

No. of servers (s)	P0	Lq	Ls	Cost of mechanics = Ls *30	Cost of servers = s * 20	Total cost per hour
2	0.3333	0.3333	1.3333	40.0	40.0	80.0
3	0.3636	0.0455	1.0455	31.4	60.0	91.4
4	0.3673	0.0068	1.0068	30.2	80.0	110.2

The table above illustrates that s = 2 is the minimum cost situation.