Q22OQ You are planning employees for a... [FREE SOLUTION]

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Operations And Supply Chain Management

F. Robert Jacobs

Business Studies

14th Edition 路 800 pages

ISBN: 9780078024023

Chapter 2: Q22OQ (page 256)

You are planning employees for a bank. You plan for six tellers. Tellers take 15 minutes per customer with a standard deviation of seven minutes. Customers will arrive one every three minutes according to an exponential distribution (recall that the standard deviation is equal to the mean). Every customer who arrives eventually gets serviced.
a. On average, how many customers would be waiting in line?
b. On average, how long would a customer spend in the bank?
c. If a customer arrived, saw the line, and decided not to get in line, that customer has .
d. A customer who enters the line but decides to leave the line before getting service is said to have .

Short Answer

Expert verified

Mean arrival time, $\overset{炉}{X_{a}} =鈥�3鈥尘颈苍$

Exponential distribution, mean and standard deviation are equal i.e. $S_{a} 鈥�=鈥�3尘颈苍$

Mean service time, $\overset{炉}{X_{a}} =鈥� 15 鈥尘颈苍$

Standard deviation, $S_{s} 鈥� = 鈥� 7 \min$

No. of tellers, S=6

Expected arrival rate

$\begin{array}{l} (位) = \frac{1}{X_{a}} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� = 鈥� \frac{1}{3} 脳 60 \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� = 20 鈥塩耻蝉迟辞尘别谤鈥塸别谤鈥塰辞耻谤 \end{array}$

Service rate per server

$\begin{array}{l} (渭) = \frac{1}{X_{s}} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� = 鈥� \frac{1}{15} 脳 60 \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� = 4 鈥塩耻蝉迟辞尘别谤鈥塸别谤鈥塰辞耻谤 \end{array}$

Coefficient of variation for arrival

$\begin{array}{l} (C_{a}) = \frac{S_{a}}{X_{a}} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌� = \frac{3}{3} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌� = 鈥� 1 \end{array}$

Coefficients of variation for service

$\begin{array}{l} (C_{s}) = \frac{S_{s}}{X_{s}} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌� = \frac{7}{15} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌� = 鈥� 0.4667 \end{array}$

Expected server utilization is

Expected server utilization

$\begin{array}{l} (蚁) =鈥� \frac{位}{厂脳鈥坝�} \end{array}$

Obtain the following results

$\begin{array}{l} (蚁) = \frac{20}{6脳4} \\ 鈥夆赌夆赌夆赌夆赌夆赌� = 鈥� 0.833 \end{array}$

The expected server utilization is 0.833

The expected number of people waiting with the help of people waiting is

Expected number of people waiting

$\begin{array}{l} L_{q} = \frac{蚁^{\sqrt{2 (s + 1)}}}{1 鈭� 蚁} 脳 \frac{C_{a}^{2} + C_{s}^{2}}{2} \\ 鈥� \\ {鈥塋}_{q} = \frac{{0.833}^{\sqrt{2 (6 + 1)}}}{1 鈭� 0.833} 脳 \frac{1^{2} + {(0.4667)}^{2}}{2} \\ 鈥夆赌� \\ 鈥夆赌夆€� = 1.8468 \end{array}$

The expected number of people waiting is 1.8468.

Step by step solution

Step by Step Solution Step 1: (a) Average number of people waiting in line is

$L_{q} = 鈥� 1.8468$

(b) Average number of people waiting in the system is

$L_{s} {=鈥塋}_{q} 鈥�+鈥塖脳蚁$

$L_{s} = 鈥�1.8468 + 6 脳 (0.8448) 鈥塩耻蝉迟辞尘别谤蝉$

Therefore, the average time waiting in the system

$\begin{array}{l} W_{s} 鈥� = 鈥� \frac{L_{s}}{位} \\ W_{s} 鈥夆赌夆€� = 鈥� \frac{6.8448}{20} 脳 60 \\ W_{s} 鈥� = 鈥�20.53鈥尘颈苍 \end{array}$

The average time waiting in the system is 20.53 min.

(c)Customer arrived, saw the line, and decided not to get in line, that customer is

If a consumer arrives, sees the queue, and decides not to join it, he is said to have balking.

(d) Enters the line but decides to leave the line before getting service is said to have is

Reneging service occurs when a customer leaves the line before the service is delivered.

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TASK	Task Time (Seconds)	Immediate Predecessor
A	30	-
B	35	A
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Short Answer

Step by step solution

Step by Step Solution Step 1: (a) Average number of people waiting in line is

(b) Average number of people waiting in the system is

(c)Customer arrived, saw the line, and decided not to get in line, that customer is

(d) Enters the line but decides to leave the line before getting service is said to have is

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