Asked by Beyza Albayrak on May 08, 2024
Verified
On a typical Monday, an average of 8 customers per hour arrive at the bank to transact business. There is one teller at the bank, and the average time required to transact business is five minutes. It is assumed that service times may be described by the exponential distribution. A single line would be used, and the customer at the front of the line would go to the first available bank teller. If a single teller is used, find:
a) The average time in the line.
b) The average number in the line.
c) The average time in the system.
d) The average number in the system.
e) The probability that the bank is empty.
f) If the business is considering adding a second teller (who would
work at the same rate as the first) to reduce the waiting time
for customers. She assumes that this will cut the waiting time in
half. If a second teller is added, find the new answers to parts (a)
to (e).
Exponential Distribution
A statistical distribution used for modeling the time between events in a process where events occur continuously and independently at a constant average rate.
Bank Teller
A financial service professional who assists customers in managing bank accounts, executing transactions such as deposits and withdrawals, and providing information on bank products.
- Harness the M/M/1 queuing model to compute system utilization, appraise waiting times, and count numbers present in the system and queue.
- Evaluate the ramifications of expanding service capacity for system performance measures.
Verified Answer
TT
tonisha thompsonMay 15, 2024
Final Answer :
a to e)
Average number of customers in the queue (Lq) B 1.333 Average number of customers in the system( Ls) D 2.000 Average waiting time in the queue (Wq) A 0.167 Average time in the system (WS) C 0.250 Probability (% of time) system is empty (Po) E 0.333\begin{array} { |l|l| }\hline \text {Average number of customers in the queue }\left(\mathrm{L}_{q}\right) & \text { B } 1.333 \\\hline \text { \(\text { Average number of customers in the system( } \left.\mathrm{L}_{\mathrm{s}}\right)\)}&\text { D } 2.000\\\hline \text {\(\text { Average waiting time in the queue }\left(\mathrm{W}_{\mathrm{q}}\right)\) }&\text { A } 0.167\\\hline \text {\(\text { Average time in the system }\left(\mathrm{W}_{\mathbf{S}}\right)\) }&\text { C } 0.250\\\hline \text {\(\text { Probability (\% of time) system is empty }(\mathrm{Po})\) }&\text { E } 0.333\\\hline\end{array}Average number of customers in the queue (Lq) Average number of customers in the system( Ls) Average waiting time in the queue (Wq) Average time in the system (WS) Probability (% of time) system is empty (Po) B 1.333 D 2.000 A 0.167 C 0.250 E 0.333
f)
Average number of customers in the queue (Lq)B0.083 Average number of customers in the system (LS)D0.750 Average waiting time in the queue (Wq)A0.010 Average time in the system (WS)C0.094 Probability (% of time) system is empty (Po)E0.500\begin{array}{|l|l|}\hline \text { Average number of customers in the queue }\left(\mathrm{L}_{\mathrm{q}}\right) & \mathrm{B} 0.083 \\\hline \text { Average number of customers in the system }\left(\mathrm{L}_{\mathrm{S}}\right) & \mathrm{D} 0.750 \\\hline \text { Average waiting time in the queue }\left(\mathrm{W}_{\mathrm{q}}\right) & \mathrm{A} 0.010 \\\hline \text { Average time in the system }\left(\mathrm{W}_{\mathrm{S}}\right) & \mathrm{C} 0.094 \\\hline \text { Probability }(\% \text { of time) system is empty }(\mathrm{Po}) & \mathrm{E} 0.500\\\hline\end{array} Average number of customers in the queue (Lq) Average number of customers in the system (LS) Average waiting time in the queue (Wq) Average time in the system (WS) Probability (% of time) system is empty (Po)B0.083D0.750A0.010C0.094E0.500
Average number of customers in the queue (Lq) B 1.333 Average number of customers in the system( Ls) D 2.000 Average waiting time in the queue (Wq) A 0.167 Average time in the system (WS) C 0.250 Probability (% of time) system is empty (Po) E 0.333\begin{array} { |l|l| }\hline \text {Average number of customers in the queue }\left(\mathrm{L}_{q}\right) & \text { B } 1.333 \\\hline \text { \(\text { Average number of customers in the system( } \left.\mathrm{L}_{\mathrm{s}}\right)\)}&\text { D } 2.000\\\hline \text {\(\text { Average waiting time in the queue }\left(\mathrm{W}_{\mathrm{q}}\right)\) }&\text { A } 0.167\\\hline \text {\(\text { Average time in the system }\left(\mathrm{W}_{\mathbf{S}}\right)\) }&\text { C } 0.250\\\hline \text {\(\text { Probability (\% of time) system is empty }(\mathrm{Po})\) }&\text { E } 0.333\\\hline\end{array}Average number of customers in the queue (Lq) Average number of customers in the system( Ls) Average waiting time in the queue (Wq) Average time in the system (WS) Probability (% of time) system is empty (Po) B 1.333 D 2.000 A 0.167 C 0.250 E 0.333
f)
Average number of customers in the queue (Lq)B0.083 Average number of customers in the system (LS)D0.750 Average waiting time in the queue (Wq)A0.010 Average time in the system (WS)C0.094 Probability (% of time) system is empty (Po)E0.500\begin{array}{|l|l|}\hline \text { Average number of customers in the queue }\left(\mathrm{L}_{\mathrm{q}}\right) & \mathrm{B} 0.083 \\\hline \text { Average number of customers in the system }\left(\mathrm{L}_{\mathrm{S}}\right) & \mathrm{D} 0.750 \\\hline \text { Average waiting time in the queue }\left(\mathrm{W}_{\mathrm{q}}\right) & \mathrm{A} 0.010 \\\hline \text { Average time in the system }\left(\mathrm{W}_{\mathrm{S}}\right) & \mathrm{C} 0.094 \\\hline \text { Probability }(\% \text { of time) system is empty }(\mathrm{Po}) & \mathrm{E} 0.500\\\hline\end{array} Average number of customers in the queue (Lq) Average number of customers in the system (LS) Average waiting time in the queue (Wq) Average time in the system (WS) Probability (% of time) system is empty (Po)B0.083D0.750A0.010C0.094E0.500
Learning Objectives
- Harness the M/M/1 queuing model to compute system utilization, appraise waiting times, and count numbers present in the system and queue.
- Evaluate the ramifications of expanding service capacity for system performance measures.