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(solution) OPERATIONS MANAGEMENT - DSC 410 FALL 2016 - WAITING LINES -


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OPERATIONS MANAGEMENT ? DSC 410

 

FALL 2016 ? WAITING LINES ? HW #1

 

DUE MONDAY OCT 10th, 2016

 

NAME:

 

1. De Morgan?s Bank is the only bank in a small town in Georgia. On a

 

typical Friday, an average of 13.6 customers arrives per hour at the

 

bank to transact business. There is one single teller at the bank, and

 

the average time required to transact business is 4 minutes. It is

 

assumed that service times are exponentially distributed. Although

 

this is the only bank in town, some people in the town have begun

 

using the bank in a neighboring town about 20 miles away. A single

 

line is used and the customer at the front of the line goes to the first

 

available bank teller. It is estimated that waiting time cost per

 

customer per hour is $25. A bank teller is paid $12 per hour

 

Arrival rate = 13.6 customers per hour

 

Service time = 4 minutes = 0.0666666 hours

 

Cost per server = $12 per hour

 

Waiting cost = $25 per hour per customer

 

This problem was run using computer software. Time units were

 

hours. The problem was run twice, first under the assumption of one

 

teller, and secondly under the assumption of 2 tellers. Below is the

 

computer output:

 

De Morgan's Bank (1 Teller)

 

QUEUE 1 : M / M / C ?[TIME UNITS = HOURS]

 

Q U E U E S TAT I S TI C S

 

Number of identical servers . . . . . . . . .

 

Mean arrival rate . . . . . . . . . . . . . .

 

Mean service rate per server . . . . . . . . 1

 

13.6000

 

15.0000 Mean server utilization (%) . . . . . . . . .

 

Expected number of customers in queue . . . .

 

Expected number of customers in system . . .

 

Probability that a customer must wait . . . .

 

Expected time in the queue . . . . . . . . .

 

Expected time in the system . . . . . . . . . 90.6667

 

8.8076

 

9.7143

 

0.9067

 

0.6476

 

0.7143 1 De Morgan's Bank (1 Teller)

 

QUEUE 1 : M / M / C [TIME UNITS = HOURS]

 

COST ANALYSIS PER UNIT TIME

 

Current System

 

Optimal System *

 

Number of servers |

 

1

 

|

 

2

 

|

 

Cost per server

 

| 12.0000

 

| 12.0000

 

|

 

Cost of service

 

|

 

12.0000|

 

24.0000|

 

Mean number in system | 9.7143

 

| 1.1412

 

|

 

Waiting cost/customer | 25.0000

 

| 25.0000

 

|

 

Cost of waiting

 

|

 

242.8600|

 

28.5300|

 

------------------TOTAL COST

 

254.8600

 

52.5300

 

* Optimization is over number of servers

 

De Morgan's Bank (1 Teller)

 

QUEUE 1 : M / M / C

 

PROBABILITY DISTRIBUTION OF NUMBER IN SYSTEM

 

Number Prob 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

 

+----+----+----+----+----+----+----+----+----+----+

 

0 0.0933|*****

 

|

 

1 0.0846|****+---|

 

2 0.0767|****--------|

 

3 0.0696|***+-----------|

 

4 0.0631|***+--------------|

 

5 0.0572|***------------------|

 

6 0.0518|***---------------------|

 

7 0.0470|**+-----------------------|

 

8 0.0426|**+-------------------------|

 

9 0.0386|**----------------------------|

 

10 0.0350|**------------------------------|

 

11 0.0318|**--------------------------------|

 

12 0.0288|*+---------------------------------|

 

13 0.0261|*+----------------------------------|

 

14 0.0237|*+-----------------------------------|

 

15 0.0215|*+-------------------------------------| 2 De Morgan's Bank (2 Tellers)

 

QUEUE 2 : M / M / C [TIME UNITS = HOURS]

 

Q U E U E S TAT I S TI C S

 

Number of identical servers . . . . . . . . .

 

Mean arrival rate . . . . . . . . . . . . . .

 

Mean service rate per server . . . . . . . . 2

 

13.6000

 

15.0000 Mean server utilization (%) . . . . . . . . .

 

Expected number of customers in queue . . . .

 

Expected number of customers in system . . .

 

Probability that a customer must wait . . . .

 

Expected time in the queue . . . . . . . . .

 

Expected time in the system . . . . . . . . . 45.3333

 

0.2345

 

1.1412

 

0.2828

 

0.0172

 

0.0839 De Morgan's Bank (2 Tellers)

 

QUEUE 2 : M / M / C [TIME UNITS = HOURS]

 

COST ANALYSIS PER UNIT TIME

 

Current System

 

Optimal System *

 

Number of servers

 

|

 

2

 

|

 

2

 

|

 

Cost per server

 

| 12.0000

 

| 12.0000

 

|

 

Cost of service

 

|

 

24.0000|

 

24.0000|

 

Mean number in system | 1.1412

 

| 1.1412

 

|

 

Waiting cost/customer

 

| 25.0000

 

| 25.0000

 

|

 

Cost of waiting

 

|

 

28.5300|

 

28.5300|

 

------------------TOTAL COST

 

52.5300

 

52.5300

 

* Optimization is over number of servers

 

De Morgan's Bank (2 Tellers)

 

QUEUE 2 : M / M / C

 

PROBABILITY DISTRIBUTION OF NUMBER IN SYSTEM

 

Number Prob 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

 

+----+----+----+----+----+----+----+----+----+----+

 

0 0.3761|*******************

 

|

 

1 0.3410|*****************+-----------------|

 

2 0.1546|********------------------------------------ |

 

3 0.0701|****------------------------------------------- |

 

4 0.0318|**-----------------------------------------------|

 

5 0.0144|*------------------------------------------------|

 

6 0.0065|+------------------------------------------------|

 

OVER 0.0055|+------------------------------------------------| 3 +----+----+----+----+----+----+----+----+----+----+ Use the above computer output to respond to the following

 

questions.

 

(a) On the average, how many minutes does a customer spend in the

 

line before getting served if one bank teller is on duty?

 

ANSWER

 

(b) On the average, how many minutes does a customer spend in the

 

line before getting served if two bank tellers are on duty?

 

ANSWER

 

(c) On the average, how many minutes does a customer spend in the

 

bank both waiting to be served and being served if one teller is on

 

duty?

 

ANSWER

 

(d)On the average, how many minutes does a customer spend in the

 

bank both waiting to be served and being served if two tellers are

 

on duty?

 

ANSWER

 

(e) On the average, how long is the waiting line if one teller is on

 

duty?

 

ANSWER (f) On the average, how long is the waiting line if two tellers are on

 

duty?

 

ANSWER 4 (g) On the average, how many customers are in the bank if one teller

 

is on duty?

 

ANSWER

 

(h) On the average, hoe many customers are in the bank if two tellers

 

are on duty?

 

ANSWER

 

(i) What percentage of the time is a teller busy if one teller is on

 

duty?

 

ANSWER

 

(j) What percentage of the time is a teller busy if two tellers are on

 

duty?

 

ANSWER

 

(k)If the bank opens for service 8 hours on Fridays, what is the

 

expected cost for both waiting and service for the day if one teller

 

is used?

 

ANSWER

 

(l) If the bank opens for service 8 hours on Fridays, what is the

 

expected cost for both waiting and service for the day if two tellers

 

are used?

 

ANSWER

 

(m) What measures of performance are cut into half when two

 

tellers are on duty as opposed to one teller on duty?

 

ANSWER 5 (n) How come other measures of performance are not cut into half

 

when we double the number of tellers?

 

ANSWER (o) What is your recommendation and why?? Should the bank

 

schedule one teller on duty or two tellers on duty?

 

ANSWER 6

 


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