Waiting Line and Queuing Theory Models
REVISED
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CHAPTER 14
WAITING LINE
AND
QUEUING THEORY MODELS
Alternative Example 14.3: A new shopping mall is considering setting up an information desk manned by two employees. Based on information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed. a. Find the proportion of the time that the employees are idle. b. Find the average number of people waiting in the system. c. Find the expected time a person spends waiting in the system. ANSWER: (servers). a. P 20/hour, 30/hour, M 2 open channels
SOLUTIONS TO DISCUSSION QUESTIONS AND PROBLEMS
14-1. The waiting line problem concerns the question of finding the ideal level of service that an organization should provide. The three components of a queuing system are arrivals, waiting line, and service facility. 14-2. The seven underlying assumptions are: 1. Arrivals are FIFO. 2. There is no balking or reneging. 3. Arrivals are independent. 4. Arrivals are Poisson. 5. Service times are independent. 6. Service times are negative exponential. 7. Average service rate exceeds average arrival rate. 14-3. The seven operating characteristics are: 1. Average number of customers in the system (L) 2. Average time spent in the system (W) 3. Average number in the queue (Lq) 4. Average time in the queue (Wq) 5. Utilization factor ( ) 6. Percent idle time (Po) 7. Probability there are more than K customers in the system
1 1 ⎛ 20 ⎞ 0 ! ⎜ 30 ⎟ ⎝ ⎠
2 3
1 2 3
0
0
1 ⎛ 20 ⎞ 1 ! ⎜ 30 ⎟ ⎝ ⎠
1
1
1 ⎛ 20 ⎞ ⎡ 2(30) ⎤ ⎥ ⎜ ⎠ ⎢ 2 ! ⎝ 30 ⎟ ⎣ 2(30) 20 ⎦
2
1
1 ⎛ 4⎞ 2 ⎜ 9⎟ ⎝ ⎠
1 3 1 2
⎡ 60 ⎤ ⎢ ⎥ ⎣ (60 20) ⎦
50%
1
b.
L
(20)(30)(20 / 30)2 ⎛ 1 ⎞ ⎜ ⎟ (1)[(2)(30) 20]2 ⎝ 2 ⎠
20 30
( 800 / 3) ⎛ 1 ⎞
⎝ ⎠ 1, 600 ⎜ 2 ⎟
L 3/ 4 20
M14_REND6289_10_IM_C14.QXD 5/12/08 1:01 PM Page 218
218
CHAPTER 14
WAITING LINE
AND
QUEUING THEORY MODELS
Alternative Example 14.3: A new shopping mall is considering setting up an information desk manned by two employees. Based on information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed. a. Find the proportion of the time that the employees are idle. b. Find the average number of people waiting in the system. c. Find the expected time a person spends waiting in the system. ANSWER: (servers). a. P 20/hour, 30/hour, M 2 open channels
SOLUTIONS TO DISCUSSION QUESTIONS AND PROBLEMS
14-1. The waiting line problem concerns the question of finding the ideal level of service that an organization should provide. The three components of a queuing system are arrivals, waiting line, and service facility. 14-2. The seven underlying assumptions are: 1. Arrivals are FIFO. 2. There is no balking or reneging. 3. Arrivals are independent. 4. Arrivals are Poisson. 5. Service times are independent. 6. Service times are negative exponential. 7. Average service rate exceeds average arrival rate. 14-3. The seven operating characteristics are: 1. Average number of customers in the system (L) 2. Average time spent in the system (W) 3. Average number in the queue (Lq) 4. Average time in the queue (Wq) 5. Utilization factor ( ) 6. Percent idle time (Po) 7. Probability there are more than K customers in the system
1 1 ⎛ 20 ⎞ 0 ! ⎜ 30 ⎟ ⎝ ⎠
2 3
1 2 3
0
0
1 ⎛ 20 ⎞ 1 ! ⎜ 30 ⎟ ⎝ ⎠
1
1
1 ⎛ 20 ⎞ ⎡ 2(30) ⎤ ⎥ ⎜ ⎠ ⎢ 2 ! ⎝ 30 ⎟ ⎣ 2(30) 20 ⎦
2
1
1 ⎛ 4⎞ 2 ⎜ 9⎟ ⎝ ⎠
1 3 1 2
⎡ 60 ⎤ ⎢ ⎥ ⎣ (60 20) ⎦
50%
1
b.
L
(20)(30)(20 / 30)2 ⎛ 1 ⎞ ⎜ ⎟ (1)[(2)(30) 20]2 ⎝ 2 ⎠
20 30
( 800 / 3) ⎛ 1 ⎞
⎝ ⎠ 1, 600 ⎜ 2 ⎟
L 3/ 4 20