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Process Analysis and Design
Operations Strategy Process Analysis Manufacturing
Service SystemDesign Matrix
Service Blueprinting
Service Classifications
Waiting Line Analysis
Services
62
Irwin/McGraw-Hill
Queues (Waiting Lines)
People waiting to be served/machines waiting to be overhauled
Can’t have inventory in services!
The issue is the trade-off between cost of service and the cost of waiting
63
Irwin/McGraw-Hill
Suggestions for Managing Queues1. Determine an acceptable waiting time
2. Divert customer’s attention while waiting
3. Inform your customers of what to expect
4. Keep other employees out of sight
5. Segment customers
6. Train employees to be friendly
7. Encourage customers during slack periods
8. Adopt a long-term perspective
63
Irwin/McGraw-Hill
Suggestions for Managing QueuesFor example, can control arrivals by:
Restricting the line (short line) - e.g., Wendy’s drive-thru
Post business hours
Establishing specific hours for specific customers or price - e.g., run specials; increase price for adult haircuts
Can also provide faster (or slower) servers, machines, layouts, set-up times, etc.
63
Irwin/McGraw-Hill
Components of the Queuing Phenomenon
CustomerArrivals
Server(s)
Waiting Line
Servicing System
Exit
PopulationSource?Infinite or
Finite
Service Rate?Constant or
Variable
Infinite Queue LengthFCFS
Balking or Reneging?
64
Irwin/McGraw-Hill
Queuing Components
Arrivals from a finite population - Limited size customer pool (important distinction, as probabilities change after each customer)
FCFS (First come, first served) - most common priority rule, but not only one - can have emergencies first, best customers first, triage, etc.
Balking (look and then leave) vs. Reneging (wait for awhile and then decide to leave)
63
Irwin/McGraw-Hill
Queuing Components (cont’d) Poisson Distribution:
- The Poisson is the most common distribution used in queuing theory for arrivals
- The Poisson distribution is discrete, as the number of arrivals must be an integer
- The probability of n arrivals within a T minute period = PT(n)=(λT)n(e-λT)/n!
63
Irwin/McGraw-Hill
Queuing Components (cont’d) Service Rates:
- The capacity of the server (in units per time pd) (e.g., a service rate of 12 completions per hour)
- Use the Exponential distribution when service times are random (as opposed to constant), where μ = avg # of customers served per time pd, and the probability the service time will be less than or equal to a time of length t: P = 1 – e-µt.
63
Irwin/McGraw-Hill
Line Structures
Single Channel
Multichannel
SinglePhase
Multiphase
One-personbarber shop
Car wash
Hospitaladmissions
Bank tellers’windows
65
(Also, “Mixed”; See text pages 251-52)
Irwin/McGraw-Hill
Properties of Waiting Line Models
Model LayoutSourcePopulation Service Pattern
1 Single channel Infinite Exponential
2 Single channel Infinite Constant
3 Multichannel Infinite Exponential
4 Single or Multi Finite Exponential
These four models share the following characteristics:· Single phase· Poisson arrival· FCFS· Unlimited queue length
66
(See formulas for each model in text on page 253)
Irwin/McGraw-Hill
Waiting Line ModelsCharacteristics of a waiting line model:
• Lq = Average number in line
• Ls = Average number in system
• Wq = Average time in line
• Ws = Average time in system
• ρ = Utilization of Server
• Pn = Probability of exactly n in system67
Irwin/McGraw-Hill
Characteris-
tics of Waiting
Lines
Definition
Model 1(single
channel,exp. service
rate)
Model 2(single
channel,constant
service rate)
Lq Average number in line 2/[(-)] 2/[2(-)]
Ls Avg. number in system /(-) Lq + (/)
Wq Average time in line/[(-)]
or Lq/ /[2(-)]
Ws Average time in system1/(-) orWq +(1/) Wq +(1/)
(“rho”) Utilization of server / -
Pn
Prob.of exactly n in system [1-(/)](/)n -
Po
Prob. of exactly zero in system 1 - (/) -
Where = Arrival rate (e.g., = 2 would represent a mean arrival rate of 2 per minute),and = Service rate or avg number of customers served per time period (e.g., 1 per 3 mins =20 per hour). and must be in the same units. (corrected 5/3/05)
Irwin/McGraw-Hill
Example: Model 1 (worked problem)
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.
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?
WP1
Irwin/McGraw-Hill
.8333 = cust/hr 30
cust/hr 25 = =
cust/hr 30 = mins) (1hr/60 mins 2
customer 1 =
cust/hr 25 =
Example: Model 1 (worked problem)
A) What is the average utilization of the employee?
B) What is the average number of customers in line?
4.167 = 25)-30(30
(25) =
) - ( L
22
q
WP2
Irwin/McGraw-Hill
Example: Model 1 (worked problem)C) What is the average number of customers in the system?
5 = 25)- (30
25 =
- =
s
L
D) What is the average waiting time in line?
mins 10 = hrs .1667 = 25)-030(3
25 =
) - ( =
qW
WP3
Irwin/McGraw-Hill
Example: Model 1 (worked problem)E) What is the average waiting time in the system?
mins 12 = hrs .2 = 25-30
1 =
-
1 =
s W
F) What is the probability that exactly two cars will bein the system (one being served and the other waiting in line)?
p = (1-n
n
)( ) p = (1- = 2
225
30
25
30)( ) .1157
WP4
Irwin/McGraw-Hill
Example: Model 2 (worked problem)
An automated pizza vending machine heats and dispenses a slice of pizza in 4 minutes.
Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution.
Determine:
A) The average number of customers in line.B) The average waiting time in the system.
WP5
Irwin/McGraw-Hill
Example: Model 2 (worked problem)
A) The average number of customers in line.
B) The average waiting time in the system.
= + = .06667 hrs +
1
15 / hr = .1333 hrs =
1 8 mins
= ( - )
= (10)
(2)(15)(15 - 10) =
2 2 2
.6667
Wq = ( - )
= )( - 10)
= .06667 hrs =
210
2 15 15(4 mins
WP6
Ws Wq
Lq
Irwin/McGraw-Hill
Approximating Customer Waiting Time
70
A “quick and dirty” method has been developed to compute average waiting time for multiple servers (see text, pp. 261-2). No assumptions about the underlying distributions are required! All that is needed are 4 numbers – the average and standard deviation of the inter-arrival time and service time.
Irwin/McGraw-Hill
Utilization and Time in System
0
30
60
90
120
150
50% 60% 70% 80% 90% 100%Utilization
Tim
e in
Sys
tem
70