Transcript Waiting Line and Queuing

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 increased
costs
 faster lines suggests an increase in service, thus an
increase in costs
 longer waiting times negatively affects customer
satisfaction

What is the ‘ideal’ level of services that a firm should
provide?
Management Uses from
Queuing Theory
 Is it worthwhile to invest effort in reducing the
service time?
 How many servers should be employed?
 Should priorities for certain types of customers
be introduced?
 Is the waiting area for customers adequate?
 Answers to these questions can be obtained with
 Analytic methods or queuing theory (formula
based); 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
computer technicians (customers arriving for service are
limited)

Infinite: ex. cars arriving at a highway tollbooth, shoppers
arriving at a supermarket (the source is forever “abundant”)
 Pattern of Arrivals

Nonrandom: arrivals take place according to some known
schedule (ex. assembly line)

Random: arrivals are independent and cannot be predicted
exactly
Random Pattern of Arrival
 Poisson Distribution

a probability distribution that can be used to
determine the probability of X transactions
arriving in a given time interval
X
P(X) = e- 
for X = 0, 1, 2, 3, 4
X!
where,
P(X) = probability of X arrivals
X = number of arrivals per unit of time
 = average arrival rate
e = 2.7183
0.25
0.25
0.20
0.20
Probability
Probability
Examples of Poisson Distribution
for Arrival Times
0.15
0.15
0.10
0.10
0.05
0.05
0 1 2 3 4 5 6 7 8 9
= 2 Distribution
X
0 1 2 3 4 5 6 7 8 9 10 11
= 4 Distribution
X
Arrival Characteristics
 Size of the calling population


Finite: ex. 300 computers on campus maintained by 5
computer technicians
Infinite: ex. Cars arriving at a highway tollbooth, shoppers
arriving at a supermarket
 Pattern of Arrivals

Random: arrivals are independent and cannot be predicted

Nonrandom: arrivals take place according to some known
exactly
schedule
 Behavior of the Arrivals


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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: switching between lines
Waiting Line Characteristics
 Queue Length

Limited: the length of the queue is limited by physical
restrictions


ex. waiting 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
Basic Single Queue Configurations
Queue
Departures
after
Service
Service
Facility
Arrivals
Single-Channel, Single-Phase System
Queue
Type 1
Service
Facility
Arrivals
Type 2
Service
Facility
Departures
after
Service
Single-Channel, Multiphase System
Queue
Arrivals
Service
Facility
1
Departures
Service
Facility
2
after
Service
Facility
3
Service
Multichannel, Single-Phase System
Queue
Type 1
Service
Facility
1
Type2
Service
Facility
1
Type 1
Service
Facility
2
Type 2
Service
Facility
2
Arrivals
Multichannel, Multiphase System
Departures
after
Service
Multiple Queue Configurations
Service
Facility
1
Multiple Queue
Service
Facility
2
Arrivals
Departures
after
Service
Facility
3
Service
Service
Facility
4
7
Take a Number
12
3
Service
Facility
1
5
Service
Facility
2
11
Arrivals
6
10
Departures
after
Service
Facility
3
Service
8
9
4
Service
Facility
4
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:

Random:
it takes the same amount of time to service each
customer or unit
service times vary across customers or units
Probability (for intervals of 1 minute)
Examples of Exponential
Distribution for Service Times
Probability (Service Takes Longer Than X Minutes) = e-uX for X > 0
u = Average Number Served per Minute
Average Service Time of 20 Minutes
Average Service Time of 1 hour
0 30 60 90 120 150 180 Service Time (Minutes)
Assumptions of the SingleChannel, 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
situations is the idea of uncertainty in


Interarrival times (arrival of customers)
Service times (service time per customer)
This means that probability and statistics are
needed 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
λ
μ-λ
Average number in system (L or Ls)
L =
Average queue length (Lq)
2
λ
Lq =
μ (μ – λ)
Average time in the system (W or Ws)
W=
1
μ-λ
Average time in queue (Wq)
Wq=
λ
μ (μ – λ)
Utilization factor ()
 =
λ
μ
Probability of idle time (P0)
P0 =
1-
λ
μ
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
μ
k+1
When to use what model?
 Use Single-channel model, when you have

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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



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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
No
Yes
Finite pop model
Constant ServTime
1 Channel
Single-Chn Model
No
Yes
Constant Model
>1 Channel
Multi-Chn Model