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Waiting Lines and Queuing Models
Queuing Theory
The study of the behavior of waiting lines Importance to business
There is a tradeoff between faster lines and increasedcosts faster lines suggests an increase in service, thus an increa
se in costs longer waiting times negatively affects customer satisfact
ion
What is the ‘ideal’ level of services that a firm should provide?
Management Uses from Queu ing Theory
Is it worthwhile to invest effort in reducing the serv ice time?
How many servers should be employed? Should priorities for certain types of customers be i
ntroduced? Is the waiting area for customers adequate?
Answers to these questions can be obtained with Analytic methods or queuing theory (formula ba
sed); and Simulation (computer based).
Queuing System Characteristics
Arrivals Waiting in Line Service Facility
Arrival Characteristics
Size of the calling population Finite: ex. 300 computers on campus maintained by 5 compute
r technicians (customers arriving for service are limited) Infinite: ex. c ars arriving at a highway tollbooth, shoppers arrivi
ng at a supermarket (the source is forever “abundant”)
Pattern of Arrivals Nonrandom: arrivals take place according to some known sche
dule (ex. assembly line) Random: arrivals are independent and cannot be predicted exa
ctly
Random Pattern of Arrival
Poisson Distribution a probability distribution that can be used to determine
the probability of X transactions arriving in a given timeinterval
P(X ) = for X = 0, 1, 2, 3, 4
where, P(X ) = probability of X arrivals X = number of arrivals per unit of time = average arrival rate e = 2.7183
e- X
X!
Examples of Poisson Distribution for Arrival Times
0 1 2 3 4 5 6 7 8 9
025
020.
015.
010.
005.
= 2 Distribution
Pro
bab
ility
Pro
bab
ility
0 1 2 3 4 5 6 7 8 9 10 11
025
020.
015.
010.
005.
4 4 444444444444
XX
Arrival Characteristics
Size of the calling population Finite: 300 5ex. computers on campus maintained by computer tec
44444444 Infinite: ex. 44444444 44 4 4444444 4444444444 44444444 44444444 44 4 ,
44444444444 Pattern of Arrivals
Random: 44444444 444 44444444444 444 444444 44 444444444 4444444 Nonrandom: ar r i val s t ake pl ace accor di ng t o some known schedul e
Behavior of the Arrivals Balking: customers who refuse to enter the system because the line is
too long Reneging: customers who enter the queue but leave without
completing their transactions Jockeying: swi t chi ng bet ween l i nes
Waiting Line Characteristics
Queue Length 444 444444 44 444 44444 44 444 4444 44 44444444 4444:
rictions ex. w aiting room
Unlimited: the length of the queue is not restricted
Queue Discipline Rule by which customers in the line are to receive service
Static: FCFS, first come first serve, FIFO, first in first out Dynamic: Priority e.g., rush jobs at a shop are processed
ahead of regular jobs
Service Facility Characteristics
Basic Queuing System Configurations Single Channel
one service provider per phase Multiple Channel
more than one service provider in a phase
Basi c Single 4444e 44444444444444
Service FacilityArrivals
Queue
Departures after Service
Single-Channel, Single-Phase System
Type 1 Service Facility
Arrivals
Queue Departures afterService
Single-Channel, Multiphase System
Type 2 Service Facility
Arrivals
Queue Departures
after
Service
Multichannel, Single-Phase System
Service Facility
1
Service Facility
2
Service Facility
3
Arrivals
Queue
Departures afterService
Multichannel, Multiphase System
Type 1 Service Facility
1
Type 1 Service Facility
2
Type2 Service Facility
1
Type 2 Service Facility
2
Multiple Queue Configurations
Arrivals
Departures
after
Service
Multiple Queue
Service Facility
1
Service Facility
2
Service Facility
3
Arrivals
Take a Number
Service Facility
4
7 3
Departures
after
Service
Service Facility
1
Service Facility
2
Service Facility
3
Service Facility
4
11
9
105
8
4
12
6
Service Facility Characteristics
Basic Queuing System Configurations Single Channel
one service provider per phase Multiple Channel
more than one service provider in a phase
Service Time Distribution Constant: it takes the same amount of time to service each
customer or unit Random: service times vary across customers or units
Examples of Exponential Distribu tion for Service Times
4444444 4444 44444440 30 60 90 120 150 180
s)
Probability (Service Takes Longer Than X Minutes) = e-uX for X > 0
Pro
bab
ility
(for
inte
rvals
of1
min
ute
)
u = Average Number Served per Minute
Average Service Time of 20 Minutes
Average Service Time of 1 hour
Assumptions of the Single-Channel, Single-Phase Model
Arrivals are served on a FIFO basis Every arrival waits to be served regardless of the length of the
line: that is there is no balking or reneging Arrivals are independent of preceding arrivals, but the
average number of arrivals (the arrival rate, λ) does not change over time
Arrivals are described by a Poisson probability distribution and come from an infinite or very large population
Service times also vary from one customer to the next and are independent of one another, but their average rate (μ) is known
Service times occur according to the negative exponential probability distribution
The average service rate is greater than the average arrival rate
Idea of Uncertainty
Note here that integral to queuing situation s is the idea of uncertainty in
Interarrival times (arrival of customers) Service times (service time per customer)
This means that probability and statistics are nee ded to analyze queuing situations.
System Performance Measures
Important to measuring the performance of the system are the parameters:
λ = the average number of arrivals per time period
μ = the average number of people or items served
per time period
System Performance Measures
Number of units in the system (customers) Average number in system (L or Ls) Average queue length (Lq)
Waiting Times Average time in the system (W or Ws) Average time in queue (Wq)
Utilization Rates Utilization factor () Probability of idle time (P0)
Queuing Equations
Average number in system (L or Ls)
Average queue length (Lq)
Average time in the system (W or Ws)
Average time in queue (Wq)
Utilization factor ()
Probability of idle time (P0)
L =
Lq =
W =
Wq=
=
P0 = 1 -
λμ - λ
λ2
μ (μ – λ)
1μ - λ
λμ (μ – λ)
λμ
λμ
Queuing Equations
Probability that the number of customers in the system is greater than k, Pn>k
where n = number of units in the system
Pn>k= ( )λ k + 1
μ
When to use what model?
Use Single-channel model, when you have Only one service provider Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying Random (inconstant) service times (Expo Dist) FIFO
When to use what model? (2)
Use Multi-channel model, when you have More than one service providers Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying Random (inconstant) service times (Expo Dist)
but both channel must perform at the same rate
When to use what model? (3)
Use Constant-service time model, when you have Constant service times (a fixed cycle) Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying
The question will be asking about either to choose the new or the old machines.
When to use what model? (4)
Use finite population model, when you have Finite source (calling population) Random (inconstant) service times (Expo Dist) Only one service providers Random pattern of arrivals (Pois Dist) FCFS
SUM
Finite Finite pop modelYes
No Constant ServTime
1 Channel >1 Channel
YesConstant Model
No
Single-Chn Model Multi-Chn Model