Question

The pharmacist at Arnold Palmer? Hospital, Wende? Huehn-Brown, receives

6 requests for prescriptions each? hour, Poisson distributed. It takes her a mean time of 5 minutes to fill? each, following a negative exponential distribution. The waiting line table is provided below and also we know that Upper W Subscript q Baseline equals StartFraction Upper L Subscript q Over lambda EndFraction

.

Table: Average Number of Customer in Waiting Line

Poisson? Arrivals, Exponential Service Times

Number of Service? Channels, M

rho

1	2	3	4	5
.10	.0111
.15	.0264	.0008
.20	.0500	.0020
.25	.0833	.0039
.30	.1285	.0069
.35	.1884	.0110
.40	.2666	.0166
.45	.3681	.0239	.0019
.50	.5000	.0333	.0030
.55	.6722	.0449	.0043
.60	.9000	.0593	.0061
.65	1.2071	.0767	.0084
.70	1.6333	.0976	.0112
.75	2.2500	.1227	.0147
.80	3.2000	.1523	.0189
.85	4.8166	.1873	.0239	.0031
.90	8.1000	.2285	.0300	.0041
.95	18.0500	.2767	.0371	.0053
1.0		.3333	.0454	.0067
1.2		.6748	.0904	.0158
1.4		1.3449	.1778	.0324	.0059
1.6		2.8444	.3128	.0604	.0121
1.8		7.6734	.5320	.1051	.0227
2.0			.8888	.1739	.0398

?a) Using the? table, the average number of prescriptions in the queue? =

?b) Using the given? relationship, the average time a prescription spends in the queue? = nothing Using the given? relationship, the average time a prescription spends in the queue after Wende has hired Ajay Agarwal? = minutes ?(round your response to two decimal? places).

Answer 1

Average arrival rate, λ = 6 per hr.
Average service rate, μ = 1 in 5 minutes = 12 per hr.

So,

(a)

λ/μ = 6/12 = 0.5

No. of servers. M = 1

From the table, the average number of prescriptions in the queue, L_q = 0.50

(b)

The average time a prescription spends in the queue, W_q = L_q / λ = 0.5/6 hrs. = 5 minutes

(c)

When Ajay was hired,

M = 2

λ/μ is still 0.5

From the table, the average number of prescriptions in the queue, L_q = 0.0333

W_q = L_q / λ = 0.0333/6 hrs. = 0.33 minutes