Queuing Models

1
Queuing Models

• Yanni Papadakis

2
Overview

• Queuing viewed from multiple perspectives
• Queuing Terminology
• Performance Metrics for Queuing Systems
• MM1 Queue
• MMk Queue
• MM1N Queue

3
Queuing Models

• PART I
• Queuing viewed from multiple perspectives

4
Economic Perspectives

• Traditional View
• Queues Hallmark of Centrally Planned Economies
• Modern View
• Queues Everywhere
• Pricing Transactions require costs that may be
  too high. Economic actors may prefer allocation
  by queuing to auctions (i.e. no queuing)
• Expect high variation in propensity to join
  queues by income (opportunity cost of waiting)

5
Economic Perspectives

• Queues act as signals
• You have hottest club in town, if you need the
  meanest bouncer and have the longest queue
• Queues raise switching costs
• Long queues outside all pubs deter barhopping
• Sometimes queues form very fast, but incentives
  take time to kick in
• Road Pricing
• Mixed allocation schemes
• People paid to wait in front of box office or
  government agency

6
Philosophical Perspectives

• Equity
• Allocation by queuing a civilized way of
  distributing precious (of vital importance)
  objects
• Transplants are distributed in a FIFO manner not
  auctions
• Queues may be bypassed if there is an obvious
  need (or if experts deem there is a need)
• Priority for disabled
• Priority in emergency room

7
Philosophical Perspectives

• Culture shapes and is shaped by queue discipline
• Queues less likely to be bypassed in more
  developed affluent regions
• Sometimes social pressure can police the queue
• When queue discipline is not obvious, there is
  room for exchanging favors
• The more networked gain over the more well off or
  more needy (depending on who a transparent queue
  discipline would favor)
• Need for transparency

8
Queuing Models

• PART II
• Queuing Terminology

9
Elements of Queuing Process

• Arrivals Customers arrive according to some
  arrival pattern.
• Queue Discipline Arriving customers may have to
  wait in one or more queues for service.
• Service Customers receive service and leave the
  system.

10
The Arrival Process

• There are two possible types of arrival processes
• Deterministic arrival process.
• Random arrival process.
• The random process is more common in businesses.
• Common interarrival times models follow
  exponential (memoryless) distribution
• Other models possible but often interarrival
  times do not depend on queue size (as with
  reservation systems)

11
Jockeying and Balking

• Jockeying occurs when customers switch lines once
  they perceived that another line is moving
  faster.
• Balking occurs if customers avoid joining the
  line when they perceive the line to be too long.

12
Tandem Queues

• These are multi-server systems.
• A customer needs to visit several service
  stations to complete the service process.
• Examples
• Patients in an emergency room.
• Passengers prepare for the next flight.
• Servers in call center

13
Homogeneity

• A homogeneous customer population is one in which
  customer needs are not known till service is
  completed.
• A non-homogeneous customer population is one in
  which customers can be categorized according to
• Different service needs
• One can utilize priority rules like shortest
  processing time

14
Queuing Models

• PART III
• Performance Metrics for Queuing Systems

15
Queue Performance Metrics

• Waiting time (average, max ,distribution)
• In queue
• In system
• Server Utilization
• Number of Customers (average, max ,distribution)
• In queue
• In system

16
Performance is measured for steady state
n
This is a steady state period..
Roughly, this is a transient period

• Initial (transient) behavior not representative
  of long run performance.

Time
17
Reaching Steady State

• EQUILIBRIUM CONDITION
• In order to achieve steady state, effective
  arrival rate must be less than sum of effective
  service rates

llt km Each with service rate of m
llt m1 m2mk For k servers with service rates mi
llt m For one server
18
Queuing Models

• PART IV
• MM1 Queue

19
MM1 Queue

• Poisson arrival process.
• Exponential service time distribution.
• A single server.
• Potentially infinite queue.

Departures at rate l
Arrivals at rate l
Service at rate m
20
The Poisson Arrival Process
(lt)ke- lt k!
P(X k)
Where l mean arrival rate per time unit t
the length of the interval e 2.7182818
(base of natural logarithms) k! k (k -1) (k
-2) (k -3) (3) (2) (1)

21
Poisson Process Excel Calculations

• We can use the POISSON function in Excel to
  determine Poisson probabilities.
• Point probability P(X k) ?
• Use Poisson(k, lt, 0)
• Example P(X 0 lt 3) POISSON(0, 1.5, 1)
• Cumulative probability P(Xk) ?
• Example P(X3 lt 3) Poisson(3, 1.5, 1)

22
Exponential Service Time
pmf f(t) me-mt
Probability service is completed before time
t P(X t) 1 - e-mt

• average number
• served per time period.
• 1/m mean service time

X t
23
Using Excel for the Exponential Probabilities

• We can use the EXPONDIST function in Excel to
  determine exponential probabilities.
• Probability density f(t) ?
• Use EXPONDIST(t, m, 0)
• Cumulative probability P(Xk) ?
• Use EXPONDIST(t, m, 1)

24
Exponential Distribution Example

• The calling center of a major insurance company
  processes claims requests over the phone. The
  service time distribution follows the exponential
  distribution with mean 5 minutes per customer.
• How many claims are processed in 5 or less
  minutes?
• How many claims are processed in 2.5 or less
  minutes?

25
Exponential Distribution Properties

• Memoryless property.
• No additional information about the time left for
  the completion of a service, is gained by
  recording the time elapsed since the service
  started
• Exponential and the Poisson distributions are
  related to one another.
• If customer arrivals follow a Poisson
  distribution with mean rate l, their interarrival
  times are exponentially distributed with mean
  time 1/l.

26
Notation

• Utilization factor ( of time server is busy)
• P0 Probability of no customers in system
• Pn Probability n customers in system
• L Average number of customers in system
• Lq Average number of customers in queue
• W Average time in system
• Wq Average time in queue

27
MM1- Performance Measures

• r l / m
• P0 1 r
• Pn rn (1 r)
• L r /(1 r)
• Lq r2 /(1 r)
• W 1 /(m l)
• Wq r /(m l)

Probability a customer is in the system for
more than t periods P(Xgtt) e-(m - l)t
28
Littles Formulas

• Littles Formulas represent important
  relationships between L, Lq, W, and Wq.
• Provided
• System is Single Queue,
• Customers arrive at a finite arrival rate l,
  and
• System operates in steady state
• L l W Lq l Wq L Lq l/m

29
Copy Room Example

• Corporation Copy Room processes 50 jobs per hour
  arriving as a Poisson process.
• Currently there are two copy machines servicing
  copy requests. Each copy machine has an
  exponential service time with mean 2 min per
  order.
• Company considers replacing them with one fast
  new copy machine operating at double the rate 1
  min per order. Fast machine costs more than two
  slow machines.
• Calculate the performance measures for the fast
  machine.

30
Queuing Models

• PART V
• MMk Queue

31
MMk Queue Characteristics

• Customers arrive according to Poisson process at
  a mean rate l.
• Service times follow exponential distribution.
• There are k servers, each works at rate m (kmgtl).

Departures rate l
Service rate m
Service rate m
Service rate m
Arrivals at rate l
32
Performance Measures
33
Performance Measures
The performance measurements W, Wq,, L are
obtained from Littles formulas.
34
Copy Room Example cont.

• l 50 arrivals/hour
• Currently there are two copy machines servicing
  copy requests. Each copy machine has an
  exponential service time with mean 2 min per
  order.
• Calculate performance measures when two slow
  machines are used.
• Is it better to have one fast machine or two
  slower ones?

35
Queuing Models

• PART V
• MM1N Queue

36
MM1N Queue

• Poisson arrival process.
• Exponential service time distribution.
• A single server.
• No more than N customers in system.

37
MM1N Queue
Other measures obtained by Littles
Law Using Average arrival rate l(1-pN)