Q23OQ You are planning the new layout ... [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: Q23OQ (page 256)

You are planning the new layout for the local branch of the Sixth Ninth Bank. You are considering separate cashier windows for the three different classes of service. Each class of service would be separate with its cashiers and customers. Oddly enough, each class of service, while different, has the same demand and service times. People for one class of service arrive every four minutes, and arrival times are exponentially distributed (The standard deviation is equal to the mean). It takes seven minutes to service each customer, and the standard deviation of the service times is three minutes. You assign two cashiers to each type of service.
On average, how long will each line be at each of the cashier windows?
On average, how long will a customer spend in the bank (assume they enter, go directly to one line, and leave as soon as service is complete)? You decide to consolidate all the cashiers so they can handle all types of customers without increasing the service times.
What will happen to the amount of time each cashier spends idle? (Increase, decrease, stay the same, depend on )
What will happen to the average amount of time a customer spends in the bank? (Increase, decrease, stay the same, depend on )

Short Answer

Expert verified

Average Wait Time (AWT), moreover known as Normal Speed of Reply (ASA), is the standard time an inbound call spends holding up in a line or holding up for a call back if that includes dynamic in your IVR framework.

Step by step solution

Step1:(a) Calculate the expected arrival rate, service coefficient of variation for arrival and service as follows:

In expected arrival rate:

$\begin{array}{l} (位) = \frac{1}{蠂_{a}} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌� = \frac{1}{4} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌�=15鈥塩ustomer鈥塸er鈥塰our \end{array}$

In service rate per server

$\begin{array}{l} (渭) = \frac{1}{蠂_{s}} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌�= \frac{1}{7} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆€�=鈥�8.57鈥塩ustomer鈥塸er鈥塰our \end{array}$

In coefficient of variation for arrival

$\begin{array}{l} C_{a} = \frac{S_{a}}{蠂_{a}} \\ 鈥夆赌夆赌夆赌夆赌夆赌�=鈥� \frac{4}{4} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌�=1 \end{array}$

In coefficient of variation for service

$\begin{array}{l} (C_{s}) = \frac{S_{s}}{蠂_{s}} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌�=鈥� \frac{3}{7} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌�=鈥�0.4285 \end{array}$

Hence, the expected arrival rate, service rate, coefficient of variation for arrival, and coefficient of service for the system are 15 customers per hour, 8.57 customers per hour, 1, and 0.4285, respectively

In the given formula:

Now, calculate the expected server utilization with the help of the given formula:

Expected server utilization

(蚁) = \frac{位}{厂脳渭}

Now putting the values in the equation, obtain the following results:

Expected server utilization is $\begin{array}{l} (蚁) = \frac{15}{2脳8.57} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆€�=0.875 \end{array}$

L_{q} 鈥�= \frac{蚁^{\sqrt{2 (s+1)}}}{1-蚁} 脳 \frac{C_{a} {鈥�+鈥嬧赌嬧赌塁}_{s}^{2}}{2} 脳 \frac{C_{a}^{2} {+鈥塁}_{s}^{2}}{2}

Now, putting the values in the equation

Obtain the following results

$\begin{array}{l} L_{q} = 鈥� \frac{{0.875}^{\sqrt{2 (2 + 1)}}}{1 鈭� 0.875} 脳 \frac{1^{2} + {(0.4285)}^{2}}{2} \\ 鈥夆赌夆赌夆赌夆赌� = 鈥� 3.4138 \end{array}$

The average number of people waiting in line

$L_{辩鈥�} =鈥�3.4138鈥夆赌塩耻蝉迟辞尘别谤$

(b) Calculate an average number of people waiting in the system with the help of the given formula

Calculate an average number of people waiting in the system with the help of the given formula:

$L_{s} {鈥�=尝}_{q} +鈥嬧€塖脳蚁$

Now putting the values in the equation, the following results:

$\begin{array}{l} L_{s} =鈥�3.4138+鈥�2脳 (0.875) \\ 鈥夆赌夆赌� = 5.1638 \end{array}$

Therefore average time waiting in the system

$\begin{array}{l} (W_{s}) = \frac{L_{s}}{位} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆€�= \frac{5.1638}{15} 鈥� \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆€�=20.655min \end{array}$

Hence, the average time waiting in the system is 20.655 min

(c) Expected server utilization is

Since the 3 lines are consolidated, arrival triples

$位=鈥�45鈥塸别谤鈥塰辞耻谤$

Number of cashiers in the system, S =6

The formulais given below:

Expected server utilization:

$(蚁) = \frac{位}{厂脳渭}$

Expected server utilization is:

role="math" localid="1651061304355" $\begin{array}{l} (蚁) = \frac{45}{6脳 (8.57)} \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€夆€�=0.875 \end{array}$

Expected server utilization is 0.875

Hence, it can be concluded that despite the change in arrival rate the amount of idle time remains the same.

(d) Standard deviation

We expect that the demand regularly disperses with a mean and standard deviation over some time. Again, remember that this approach considers as it were the likelihood of running out of stock, not how many units we are brief. To decide the possibility of stocking out over the time, we can essentially plot a typical dispersion for the anticipated request and note where the sum we have on hand lies on the bend

The average amount of time a customer spends in the bank

Hence, it can be concluded that the new average amount spent by a customer decreases to 10.83 after a change in the value of $位$ and $L_{q}$ .

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Short Answer

Step by step solution

Step1:(a) Calculate the expected arrival rate, service coefficient of variation for arrival and service as follows:

(b) Calculate an average number of people waiting in the system with the help of the given formula

(c) Expected server utilization is

(d) Standard deviation

The average amount of time a customer spends in the bank

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