### B.8 Waiting Line Economic Analysis

```Readings
Chapter 11, Sections 1, 2, 3, 5
Waiting Line Models
BA 452 Lesson B.8 Waiting Line Economic Analysis
1
Overview
Overview
BA 452 Lesson B.8 Waiting Line Economic Analysis
2
Overview
Analytical Formulas for Multiple Channels for operating characteristics have
been derived, under the queue discipline first-come, first-served, for several
queuing models with multiple channels.
M/M/2 Queuing System designates M = Markov (memoryless) arrival
distribution (Poisson), M = Markov service-time distribution (exponential), and 2
service channels, under first-come, first-served.
Economic Analysis of waiting lines maximizes profit for a firm by maximizing
value for customers. Maximizing customer value trades off quick service with
low purchase prices (resulting from lower costs).
Economic Analysis with Teamwork maximizes firm’s profits and customer’s
value by trading off having more service channels with teams providing faster
service in each channel.
BA 452 Lesson B.8 Waiting Line Economic Analysis
3
Overview
Tool Summary
Use analytical formulas or Management Scientist to compute
performance:
1) Probability that no units are in the system: P0
2) Average number of units in waiting line: Lq
3) Average number of units in system: L = Lq + l/m
4) Average time a unit spends in waiting line: Wq = Lq/l
5) Average time a unit spends in the system: W = Wq + 1/m
6) Probability that an arriving unit has to wait for service: Pw
7) Probability of n units in the system: Pn
Compute total hourly cost for units in the system
= (\$ waiting cost per hour) x (Average number of units in system)
Note average number of units in system is the only right choice above;
average time in waiting line does not count the number of units.
BA 452 Lesson B.8 Waiting Line Economic Analysis
4
Analytical Formulas for Multiple Channels
Analytical Formulas for Multiple
Channels
BA 452 Lesson B.8 Waiting Line Economic Analysis
5
Analytical Formulas for Multiple Channels
Overview
Analytical Formulas for Multiple Channels, for M/M/k
under FCFS require assumptions, some of which are
not 100% realistic:
 Multiple channels (with one central waiting line)
 Poisson arrival-rate distribution
 Exponential service-time distribution
 Unlimited maximum queue (waiting line) length
 Examples:
• Four-teller transaction counter in bank
• Two-clerk returns counter in retail store
BA 452 Lesson B.8 Waiting Line Economic Analysis
6
Analytical Formulas for Multiple Channels
1) Probability that no units are in the system:
P0 
1
n
k  1 (l
/ m)
n 0
n!


(l / m )
k!
k
(
km
km  l
)
2) Average number of units in waiting line:
Lq 
lm (l m )
k
( k  1)!( k m  l )
2
( P0 ) 
(30)(30)(30 30)
(1!)(2(30)  30)
2
2
(1/3) 
1
3
3) Average number of units in system: L = Lq + l/m
4) Average time a unit spends in waiting line: Wq = Lq/l
5) Average time a unit spends in the system: W = Wq + 1/m
BA 452 Lesson B.8 Waiting Line Economic Analysis
7
Analytical Formulas for Multiple Channels
6) Probability that an arriving unit has to wait for service: Pw
= (1/k!) (l/m)k (km/(kml)) P0
7) Probability of n units in the system:
[ (l/m)n /n! ] P0
for n < k
Pn =
[ (l/m)n /(k! k(n-k)) ] P0 for n > k
BA 452 Lesson B.8 Waiting Line Economic Analysis
8
M/M/2 Queuing System
M/M/2 Queuing System
BA 452 Lesson B.8 Waiting Line Economic Analysis
9
M/M/2 Queuing System
Overview
M/M/2 Queuing System designates M = Markov
(memoryless) arrival distribution (exponential), M =
Markov service-time distribution (Poisson), and 2
service channels, under first-come, first-served.
BA 452 Lesson B.8 Waiting Line Economic Analysis
10
M/M/2 Queuing System
Question: Smith, Jones, Johnson, and Thomas, Inc. has
begun a major advertising campaign which it believes
will increase its business 50%. To handle the increased
volume, the company has hired an additional floor
trader, Fred Hanson, who works at the same speed as
Joe Ferris.
Note that the new arrival rate of orders, l, is 50% higher
than that of Example 1 in Lesson 2.5. Thus, l = 1.5(20)
= 30 per hour.
BA 452 Lesson B.8 Waiting Line Economic Analysis
11
M/M/2 Queuing System



M/M/1:
P0 = 1-l/m
Lq = l2/(m(ml))
L = Lq + l/m
Wq = Lq/l
W = 1/(ml)
Pw = l/m
Pn = (l/m)nP0
Sufficient Service Rate
Will Joe Ferris alone be able to handle the
increase in orders?
Answer: Since Joe Ferris processes orders at a
mean rate of µ = 30 per hour, then l = µ = 30 and the
average time a unit spends in the system is
W = 1/(ml) = 1/0 = infinity.
That implies the queue of orders will grow infinitely
large. Hence, Joe alone cannot handle that increase
in demand.
BA 452 Lesson B.8 Waiting Line Economic Analysis
12
M/M/2 Queuing System



Probability of n Units in System
What is the probability that neither Joe nor Fred will be
working on an order at any point in time?
Answer: This is an M/M/k queue with l = 30 per hour, m
= 30 per hour, and k = 2. The probability that neither
Joe nor Fred will be working = the probability of no units
in the system. Analytical Formula #1 says that is:
P0 
1
n
k  1 (l
/ m)
n 0
n!


(l / m )
k!
k
(
km
km  l
)
= 1/[(1 + (1/1!)(30/30)1] + [(1/2!)(1)2][2(30)/(2(30)-30)]
= 1/(1 + 1 + 1) = 1/3 =
.333
BA 452 Lesson B.8 Waiting Line Economic Analysis
13
M/M/2 Queuing System



Average Time in System
What is the average turnaround time for an order with
both Joe and Fred working?
Answer: The average turnaround time = the average
time a unit spends in the system, W. Analytical
Formula #2 and 3 say
k
2
lm (l m )
(30)(30)(30 30)
1
Lq 
( k  1)!( k m  l )
2
( P0 ) 
(1!)(2(30)  30)
2
(1/3) 
3
and L = Lq + (l /µ) = 1/3 + (30/30) = 4/3. Finally,
W = L/l (4/3)/30 = 4/90 hr. = 2.67 min.
BA 452 Lesson B.8 Waiting Line Economic Analysis
14
M/M/2 Queuing System



Average length of queue
What is the average number of orders waiting to be
filled with both Joe and Fred working?
Answer: The average number of orders waiting to be
filled = the average number of units in the waiting
line, Lq. That was calculated earlier as 1/3.
BA 452 Lesson B.8 Waiting Line Economic Analysis
15
M/M/2 Queuing System
BA 452 Lesson B.8 Waiting Line Economic Analysis
16
M/M/2 Queuing System
BA 452 Lesson B.8 Waiting Line Economic Analysis
17
Economic Analysis
Economic Analysis
BA 452 Lesson B.8 Waiting Line Economic Analysis
18
Economic Analysis
Overview
Economic Analysis of waiting lines maximizes profit for a
firm by maximizing value for customers. Maximizing
customer value trades off quick service with low purchase
prices (resulting from the lower costs of having fewer or
less qualified employees). Wealthy customers (like at
Malibu Yogurt) prefer quick service, even if that means
higher purchase prices to pay for more or better
employees, but poor customers (like at Popeye's Chicken
in Oxnard) prefer lower purchase prices, even if that means
slower service from fewer or incompetent employees.
BA 452 Lesson B.8 Waiting Line Economic Analysis
19
Economic Analysis
Question: The advertising campaign of Smith, Jones,
Johnson and Thomas, Inc. was so successful that business
doubled. The mean rate of stock orders arriving at the
exchange is now 40 per hour and the company must
hired can process an order in an average time of 2
minutes. (So far, l = 40/hr. and m = 30/hr.)
BA 452 Lesson B.8 Waiting Line Economic Analysis
20
Economic Analysis
The brokerage firm has determined the average
waiting cost per minute for an order to be \$.50. (So, you
can charge \$.50 more per order if you can process it an
average of 1 minute faster.) Floor traders hired will earn
\$20 per hour in wages and benefits. Hence, compare
the total hourly cost of hiring 2 traders with that of hiring
= (Total salary cost per hour)
+ (Total hourly cost for orders in the system)
+ (\$30 waiting cost per hour) x (Average number
of orders in system)
= 20k + 30L.
BA 452 Lesson B.8 Waiting Line Economic Analysis
21
Economic Analysis

This is an M/M/2 queue with l = 40 per hour
m = 30 per hour. Analytical Formulae #1, 2, 3
P0 
l = 40/hr.
and
m = 30/hr.
Cost = 20k + 30L
1
n
k  1 (l
/ m)
n 0
n!


(l / m )
k!
k
(
km
km  l
)
P0 = 1 / [1+(1/1!)(40/30)]+[(1/2!)(40/30)2(60/(60-40))]
= 1 / [1 + (4/3) + (8/3)] = 1/5
Lq 
lm (l m )
k
( k  1)!( k m  l )
2
( P0 ) 
(40)(30)(40 30)
(1!)(2(30)  40)
2
2
(1/5) 
16
15
say the average number of units in the system is:
L = Lq + (l /µ) = 16/15 + 4/3 = 2.40
Hence, total cost = (20)(2) + 30(2.40) = \$112.00 per hour
BA 452 Lesson B.8 Waiting Line Economic Analysis
22
Economic Analysis

This is an M/M/3 queue with l = 40 per hour
m = 30 per hour. Analytical Formulae #1, 2, 3
P0 
l = 40/hr.
and
m = 30/hr.
Cost = 20k + 30L
1
n
k  1 (l
/ m)
n 0
n!


(l / m )
k!
k
(
km
km  l
)
P0 = 1/[[1+(1/1!)(40/30)+(1/2!)(40/30)2]+
[(1/3!)(40/30)3(90/(90-40))] ]
= 1 / [1 + 4/3 + 8/9 + 32/45] = 15/59
Lq 
lm (l m )
k
( k  1)!( k m  l )
2
( P0 ) 
(30)(40)(40 30)
(2!)(3(30)  40)
3
2
(15/59)  .1446
say the average number of units in the system is:
L = .1446 + 40/30 = 1.4780
Hence, total cost = (20)(3) + 30(1.4780) = \$104.35 per hour
BA 452 Lesson B.8 Waiting Line Economic Analysis
23
Economic Analysis

System cost comparison
Wage
Cost/Hr
\$40.00
\$60.00
Waiting
Cost/Hr
\$72.00
\$44.35
Total
Cost/Hr
\$112.00
\$104.35
Thus, the total cost of having 3 traders is less than
BA 452 Lesson B.8 Waiting Line Economic Analysis
24
Economic Analysis with Teamwork
Economic Analysis with Teamwork
BA 452 Lesson B.8 Waiting Line Economic Analysis
25
Economic Analysis with Teamwork
Overview
Economic Analysis with Teamwork maximizes firm’s profits
and customer’s value by trading off having more service
channels (like registers in a grocery) with teams of workers
(like a cashier and bagger working at the same register in a
grocery) providing faster service in each channel.
BA 452 Lesson B.8 Waiting Line Economic Analysis
26
Economic Analysis with Teamwork





A fast-food franchise is considering adding a
drive-up window to a particular location.
Assume customer arrivals follow a Poisson probability
distribution, with an arrival rate of l = 24 cars per hour.
Assume customer service times follow an exponential
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.
BA 452 Lesson B.8 Waiting Line Economic Analysis
27
Economic Analysis with Teamwork



System A. One employee fills the order and
takes the money from the customer. The average
service time for this alternative is 2 minutes (30
customers per hour). All together, (k, l, m) = (1, 24, 30).
System B. 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
(48 customers per hour). All together,
(k, l, m) = (1, 24, 48).
System C. Two service windows, each with an employee
that fills the order and takes the money from the
customer. The average service time for this alternative
is 2 minutes (30 customers per hour) for each channel.
All together, (k, l, m) = (2, 24, 30).
BA 452 Lesson B.8 Waiting Line Economic Analysis
28
Economic Analysis with Teamwork




Customer waiting time is valued at \$25 per hour.
 So, you can charge \$25/60 more per order if you can
process orders an average of 1 minute (1/60 hour)
faster.
The cost of each employee is \$6.50 per hour.
Each channel costs \$20 for equipment and space.
Which system is most profitable?
BA 452 Lesson B.8 Waiting Line Economic Analysis
29
Economic Analysis with Teamwork

The labor plus equipment-space cost for each
System A:
6.50
+
20.00
=
\$26.50/hour
B:
2(6.50)
+
20.00
=
\$33.00/hour
channel of each system is:System
System C:
6.50
+
20.00
=
\$26.50/hour

The waiting plus channel cost for each each system is:
System A:
System B:
System C:

25(4)
25(1)
25(0.9524)
+
+
+
26.50(1)
33.00(1)
26.50(2)
=
=
=
\$126.50
\$ 58.00
\$ 76.81
System B is thus most profitable (it costs the minimum,
\$58.00). That is, one employee fills the order while a
second employee takes the money from the customer.
BA 452 Lesson B.8 Waiting Line Economic Analysis
30
Economic Analysis with Teamwork




If, instead, the drive-up window were for a
location in a poor part of town, where customer waiting
time is valued at \$2 per hour, which system is most
profitable?
Answer: Change the waiting plus channel cost for each
System A:
25(4)
+
26.50(1)
=
\$126.50
System
B:
25(1)
+
33.00(1)
=
system from System C: 25(0.9524) + 26.50(2) = \$\$ 58.00
to reduce time
76.81
value from \$25 to \$2. System A is now most profitable.
Finally, if the drive-up window were for outside the
colony in Malibu, where customer waiting time is valued
at \$200 per hour, which system is most profitable?
Answer: System B is most profitable, but if the value of
time were high enough, then System C would be most
profitable.
BA 452 Lesson B.8 Waiting Line Economic Analysis
31
BA 452
Quantitative Analysis
End of Lesson B.8
BA 452 Lesson B.8 Waiting Line Economic Analysis
32
```