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1

• Waiting Line Characteristics

• Suggestions for Managing Queues

• Examples (Models 1, 2, 3, and 4)

Ch 6 Tech Notes OBJECTIVES

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2

Components of a Queuing System

Exhibit TN 6.3

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3

Arrival and Service Profiles

Exhibit TN 6.2

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4

Exponential Distribution

Exhibit TN 6.4

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5

Poisson Distribution

Exhibit TN 6.5

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Customer Arrivals in Queues

Exhibit TN 6.6

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Line Structures

Exhibit TN 6.7

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Notation: Infinite Queuing: Models 1-3

linein tingnumber wai Average

server single afor

rate sevice torate arrival totalof Ratio = =

arrivalsbetween timeAverage

timeservice Average

rate Service =

rate Arrival =

1

1

qL

linein tingnumber wai Average

server single afor

rate sevice torate arrival totalof Ratio = =

arrivalsbetween timeAverage

timeservice Average

rate Service =

rate Arrival =

1

1

qL

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9Infinite Queuing Models 1-3 (Continued)

linein waitingofy Probabilit

systemin units exactly ofy Probabilit

channels service identical ofNumber =

system in the units ofNumber

served) be to time(including

systemin time totalAverage

linein waiting timeAverage =

served) being those(including

systemin number Average =

q

s

w

n

s

P

nP

S

n

W

W

L

linein waitingofy Probabilit

systemin units exactly ofy Probabilit

channels service identical ofNumber =

system in the units ofNumber

served) be to time(including

systemin time totalAverage

linein waiting timeAverage =

served) being those(including

systemin number Average =

q

s

w

n

s

P

nP

S

n

W

W

L

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Example: Model 1Assume a drive-up window at a fast food restaurant.Customers arrive at the rate of 25 per hour.The employee can serve one customer every two minutes.Assume Poisson arrival and exponential service rates.

Determine:A) What is the average utilization of the employee?B) What is the average number of customers in line?C) What is the average number of customers in the system?D) What is the average waiting time in line?E) What is the average waiting time in the system?F) What is the probability that exactly two cars will be in the system?

Determine:A) What is the average utilization of the employee?B) What is the average number of customers in line?C) What is the average number of customers in the system?D) What is the average waiting time in line?E) What is the average waiting time in the system?F) What is the probability that exactly two cars will be in the system?

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= 25 cust / hr

= 1 customer

2 mins (1hr / 60 mins) = 30 cust / hr

= = 25 cust / hr

30 cust / hr = .8333

= 25 cust / hr

= 1 customer

2 mins (1hr / 60 mins) = 30 cust / hr

= = 25 cust / hr

30 cust / hr = .8333

Example: Model 1

A) What is the average utilization of the employee?

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Example: Model 1

B) What is the average number of customers in line?

4.167 = 25)-30(30

(25) =

) - ( =

22

qL 4.167 = 25)-30(30

(25) =

) - ( =

22

qL

C) What is the average number of customers in the system?

5 = 25)-(30

25 =

- =

sL 5 = 25)-(30

25 =

- =

sL

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Example: Model 1

D) What is the average waiting time in line?

mins 10 = hrs .1667 =

=q

q

LW mins 10 = hrs .1667 =

=q

q

LW

E) What is the average waiting time in the system?

mins 12 = hrs .2 = =

ss

LW mins 12 = hrs .2 = =

s

s

LW

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Example: Model 1

F) What is the probability that exactly two cars will be in the system (one being served and the other waiting in line)?

p = (1-n

n

)( )p = (1-n

n

)( )

p = (1- = 2

225

30

25

30)( ) .1157p = (1- =

2

225

30

25

30)( ) .1157

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Little’s Law

• For nearly all queuing systems, there is a relationship between the expected time a unit spends in the system or queue and the expected number of units in the system or queue. Known as Little's flow equations, they are:

Ls = Ws and Lq = Wq

(Anderson, Sweeney, & Williams)

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Economic Analysis

• Total Cost = (cost of providing service) + (cost of waiting)

• TC = csk + cq(/)Wq = csk + cqWs

= css + cqLs, by Little’s Law, where

cs is the cost of service per channel

cq Is the cost for one customer to wait one unit of time

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Economic Analysis, suggested problem (Anderson, Sweeney & Williams)

A fast food franchise is considering operating a drive-up window food-service operation. Assume that customer arrivals follow a Poisson probability distribution, with a mean arrival rate of 24 cars per hour, and that service times follow an exponential probability distribution. Arriving customers place orders at an intercom station at the back of the parking lot and then drive to the service window to pay for and receive their orders. The following three service alternatives are being considered:

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Economic Analysis, suggested problem

• A single-channel operation in which one employee fills the order and takes the money from the customer. The average service time for this alternative is 2 minutes.

• A single-channel operation in which one employee fills the order while a second employee takes the money from the customer. The average service time for this alternative is 1.25 minutes.

• A two-channel operation with two service windows and two employees. The employee stationed at each window fills the order and takes the money for customers ar riving at the window. The average service time for this alternative is 2 minutes for each channel.

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For the suggested problem, the following cost

information is available for the fast food franchise

• Customer waiting time is valued at $25 per hour to reflect the fact that waiting time is costly to the fast food business.

• The cost of each employee is $6.50 per hour.• To account for equipment and space, an additional cost

of $20 per hour is attributable to each channel.

What is the lowest-cost design for the fast food business?