Question

Consider a bank branch that has three distinct customer arrival patterns throughout the day, as measured by average arrival rates (below) Morning (8:30 11:30): 40/hour Lunch (1:30-1:30): 60/hour Afternoon (1:30 4:00): 3 35/hour. Regardless of the time of day, the average time it takes for a teller to serve customers is 2 minutes, 52 seconds Because of competition with other banks in the area, management has determined that they want to keep the average customer wait before service to be less than 7 minutes. With that in mind, answer the following (1.) During the morning period, what is the minimum number of tellers that the bank needs to hire to achieve the 7 minute service objective mentioned above? O A. 2 tellers O B. 3 tellers ° C. The bank can't hire enough tellers to achieve an average wait before service of under 7 minutes O D. 4 tellers (2.) During lunch, what is the minimum number of tellers that the bank needs to hire to achieve the 7 minute service objective mentioned above? O A. 5 tellers O B. 3 tellers O C. 4 tellers O D. Any number of tellers will suffice to meet the objective (3.) In the afternoon, what is the minimum number of tellers that the bank needs to hire to achieve the 7 minute service objective mentioned above? O A. 3 tellers B. 1 teller C 2 tellers

Answer 1

Existing customer arrival pattern during different times in a day is as follows: A B C Arrival rate a (per hour) (per minutes) 0.667 60 1.000 35 0.583 2 Shift 3 Morning 4 Lunch 5 Afternoon D __ Service rate u (customer per minute) 2 min 52 seconds or 2.87 minutes 40 0.348 Table 1: Value of arrival and service rate In this case, Wa (the average waiting time spent by the teller in the queue) is calculated first at different number of servers (tellers). Now, number of servers at which Wa is obtained nearer to 7-minutes will be chosen as ideal number of servers for the particular shift. Use the following formulas step-wise to calculate for minimum number of tellers required to be hired to achieve the 7-minute service objective. Step 1: Calculate system utilization as follows: Step 2: Calculate Lg. Refer table "Infinite source values for Lgand Pogiven p and M”. Step 3: Calculate the average waiting time (W) as follows:

(1.) Calculate the minimum number of tellers that the bank needs to achieve the 7-minute in the morning shift service objective as follows: Step 1: Calculate system utilization as follows: (Refer table 1 for the values of Arrival and Service rate) 0.667 p= 0.348 p=1.916 p=1.9 Step 2: Calculate Lg at different number of servers. Refer table “Infinite source values for Lq and Po given p and M”.

λμ M L, P, 1.1 2 290 0.477 0.066 0.011 0.675 0.094 0.016 0.003 0.951 0.130 0.023 0.004 1.345 264 271 272 .176 236 0.177 245 .143 211 0.032 0.006 1.929 0.237 0.045 0.009 2.844 0.313 0.060 0.012 4.426 0.409 221 .223 .111 187 199 201 4 .081 166 180 .182 0.000 1.9 0.017 7.674 0.532 0.105 0.023 17.587 0.688 0.136 0.030 0.007 .053 .146 162 .165 .026 .128 .145 149 149

Thus, Number of servers 3 | 17.587 0.688 0.136 0.03 Step 3: Calculate the average waiting time (W) at different number of servers and at different values of Lq using formula given as follows: - No P Morning shift Number of servers La a ] W, 17.587 0.667 =N4/04 0.688 0.667 =N5/05 0.136 0.667 =N6/06 0.03 0.667 =N7/07

M O P N Morning shift Number ofservers 回「回」四 17.587 0.688 0.136 0.03 0.667 0.667 0.667 0.667 26.3805 1.032 0.204 0.045 Here, for 2 number of servers (M) value of Wa is 26.3805 which is greater than 7-minutes objective service time. While at 3 number servers it is 1.032 and when number of servers are 4, it is 0.204. Thus, the minimum number of tellers in the morning shift that the bank needs to achieve the 7-minute service objective 3. (2.) Calculate the minimum number of tellers during lunch shift that the bank needs to achieve the 7-minute service objective as follows: Step 1: Calculate system utilization as follows: (Refer table 1 for the values of Arrival and Service rate) p=1 0.348 p=2.874 p=2.9

Step 2: Calculate L, at different number of servers. Refer table “Infinite source values for Lq and P. given p and M”. NM M L P. .056 .083 .089 .090 .091 .045 .074 .080 .082 .082 .035 .065 .072 .074 .074 2.589 0.431 0.105 0.027 0.007 3.511 0.533 0.130 0.034 0.009 4.933 0.658 0.161 0.043 0.011 7.354 0.811 0.198 0.053 0.014 12.273 1.000 0.241 0.066 0.018 27.193 1.234 0.293 0.081 0.023 .025 .057 .065 .067 .067 .016 .050 .058 .060 061 2008 .044 .052 .054 .055

Thus, Number of servers 4 5 27.193 1.234 0.293 0.081 0.023 Step 3: Calculate the average waiting time (W) at different number of servers and at different values of Lq using formula given as follows: K M N 1 L Lunch shift Number of servers 27.193 1.234 0.293 0.081 0.023 =L4/ M4 =L5/M5 =L6/M6 I=L7/M7 L8/M8

M N L Lunch shift Number of servers q a w 27.193 1.234 0.293 | 0.081 | 0.023 1.00 27.193 1.00 1.234 1.00 0.293 1.000.081 1.00 0.023 Here, for 3 number of servers (M) value of Wa is 27.193 which is greater than 7-minutes objective service time. While at 4 number of servers it is 1.234, at 5 and 6 number of servers it is 0.293 and 0.081 respectively and when number of servers are 7, it is 0.023. Thus, the minimum number of tellers in the lunch shift that the bank needs to achieve the 7-minute service objective 4.

(3.) Calculate the minimum number of tellers during afternoon shift that the bank needs to achieve the 7-minute service objective as follows: Step 1: Calculate system utilization as follows: (Refer table 1 for the values of Arrival and Service rate) 0.583 P=0.348 p=1.676 p=1.7 Step 2: Calculate L, at different number of servers. Refer table "Infinite source values for L and Po given p and M".

Nu ML, Po 1.1 290 .294 .300 WNWNWNWN .301 .212 264 4 0.477 0.066 0.011 0.675 0.094 0.016 0.003 0.951 0.130 0.023 0.004 1.345 0.177 0.032 0.006 1.929 0.237 0.045 0.009 2.844 0.313 0.060 0.012 4.426 0.409 0.080 0.017 .271 .272 .176 .236 .245 .246 .143 .211 221 WNWNWN .111 .187 .199 .201 .081 .166 .180 .182 5 Thus, Number of servers 4.426 0.409 0.08 0.0017 Step 3: Calculate the average waiting time (w) at different number of servers and at different values of L, using formula given as follows:

L M Afternoon shift N Number of servers L, 2 W 4.426 0.583 0.409 0.583 0.08 0.583 0.0017 0.583 =L4/M4 =L5/M5 =L6/M6 =L7/M7 M N L Afternoon shift Number of servers © a w 2 3 4.426 | 0.409 0.08 0.0017 0.583 0.583 0.583 0.583 7.592 0.702 0.137 0.003 Here, for 2 number of servers (M) value of W.is 7.592 which is greater than 7-minutes objective service time. While at 3 number of servers it is 0.702, and at 4 and 5 number of servers, it is 0.0137 and 0.003 respectively. Thus, the minimum number of tellers in the afternoon shift that the bank needs to achieve the before 7-minute service objective 3.

(4.) Calculate the average number of customers waiting before they get served during afternoon, when 3 tellers are hired, as follows: Step 1: Calculate system utilization as follows: (Refer table 1 for the values of Arrival and Service rate) 0.583 0.348 p=1.676 p=1.7 Step 2: Calculate L, at 3 number of servers. Refer table "Infinite source values for L and Po given p and M".

Nu ML, Po 1.1 2 SES 236 0.477 290 0.066 327 0.011 332 0.675 250 0.094 0.016 0.003 301 0.951 212 0.130 .264 0.023 0.004 1.345 .176 0.177 0.032 245 0.006 246 1.929 .143 0.237 0.045 .221 0.009 .223 2.844 .111 0.313 .187 0.060199 0.012201 4.426 .081 0.409 .166 0.080 180 0.017 .182 3 4 5 This provide corresponding value of L, as 0.409. Step 3: Calculate the average waiting time (w.) at 3 number of servers and at calculated values of L, using formula given as follows:

0.409 0.583 W, = 0.702 W, = 0.7 Among the given options, this calculated value of W is nothing but the rounded value for 0.65 customers. NOTE: If one goes for another computational method, one can get the exact answer, but after rounding off to one decimal place answer will remain same, i.e. 0.7 Thus, correct option is B 0.65 customers (or 0.7 customers).