# Waiting Line and Queuing Theory Models

**Topics:**Queueing theory, Queue area, Cutting

**Pages:**33 (4902 words)

**Published:**April 24, 2013

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

2 3

1 12

8 12

9 12

3 people p 4

14-4. If the service rate is not greater than the arrival rate, an inﬁnite queue will eventually build up. 14-5. First-in, ﬁrst-out (FIFO) is often not applicable. Some examples are (1) hospital emergency rooms, (2) an elevator, (3) an airplane trip, (4) a small store where the shopkeeper serves whoever can get his or her attention ﬁrst, (5) a computer system set to accept priority runs, (6) a college registration system that allows juniors and seniors to register ahead of freshmen and sophomores, (7) a restaurant that may seat a party of 2 before a party of 4 even though the latter group arrived earlier, (8) a garage that repairs cars with minor problems before it works on major overhauls. 14-6. Examples of ﬁnite queuing situations include (1) a ﬁrm that has only 3 or 4 machines that need servicing, (2) a small airport at which only 10 or 15 ﬂights land each day, (3) a classroom that seats only 30 students for class, (4) a physician who has a limited number of patients, and (5) a hospital ward with only 20 patients who need care. 14-7. a. Barbershop: usually a single-channel, multipleservice system (if there is more than one barber). Arrivals Waiting line Service customers wanting haircuts seated customers who informally recognize who arrived ﬁrst among them haircut, style, shampoo, and so forth; if service involves barber, then shampooist, then manicurist, it becomes a multiphase system

c.

W

3 hr. 80

0.0375

Alternative Example 14.4: Three students arrive per minute at a coffee machine that dispenses exactly 4 cups/minute at a constant rate. Describe the operating system parameters. ANSWER: 3/minute 2

Lq

2 (

)

4/minute 9 2( 4)( 4 3)

Wq

1.125 people in queue on average 3 2 ( ) 2( 4)( 4 3)

0.375 minutes in the queue waiting 3 L Lq 1.125 4 1.87 people in the system 1 1 Wq .375 4 0.625 minutes in the system

W

REVISED

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