Queuing Theory

1
Queuing Theory

• Chapter 10

2
Why Study Queues?

• Standing in Line Costs
• We loose customers
• We waste employee time
• We bottleneck production processes
• Eliminating Lines Costs
• We have to have more servers or shorter service
  times
• So whats the best trade-off?

3
Components of Queuing Process

• Arrivals (Random)
• Servers (Random)
• Waiting Lines or Queues
• Discipline First Come, First Served (FCFS),
  LCFS, etc.

4
Basic Structures

• Channel ServersPhase Steps in Service
• Single Channel, Single Phase
• Multiple Channel, Single Phase
• Single Channel, Multiple Phase
• Multiple Channel, Multiple Phase
• Other

5
Costs of a Queue
Total Cost
cost

Service Costs
Waiting Costs
Level of Service
6
Operating Characteristics

• Probability of the Service being idle
• No units in the system P0
• Probability of some specified number of customers
  (units) are in the system
• Pn
• Mean (expected) waiting time for each customer (W
  for total or Wq in queue)

7

• Mean (expected) number in the system
• L
• Mean (expected) number in the queue
• Lq

8
Model Assumptions

• Arrival Distribution
• Most Commonly a Poisson Distribution
• Discrete Distribution (See page 474)

Where r Number of arrivals P(r) Probability
of r arrivals l Mean arrival rate e 2.71828
(base of natural log) r! r factoral ( r )
(r-1)(r-2)(r-3)..
P(r) e - l (l)r r!
9
Mean Time Between Arrives(More Assumptions)

• Negative exponential distribution with mean of 1
  / l
• Example If mean arrival rate is 2 per hour then
  the mean time between arrivals is 1 / l or 1 / 2
  hours or 30 minutes

10
Distribution of Service Times(More Assumptions)

• Most commonly negative exponential probability
  density function
• Area under the curve (See page 475)

When t Service Times f(t) probability
density function t m mean service rate 1 / m
mean service time e 2.71828 (base of natural
log)
f(t) m e -m t
11
Other Issues

• Infinite vs. Finite Calling Population
• Infinite vs. Finite Queue Length
• Steady State vs. Transient System
• Arrival Rate vs. Service Rate
• Ratio of l / m will be less then one or the
  queue expands uncontrollably
• Exponential relationship (see page 478)

12
Queuing Notation

• Written(a/b/c) (d/e/f)
• a arrival dist
• b service dist
• c of servers
• d queue discipline
• e max in queue
• f size of calling pop

• Distributions
• M Poisson
• D Deterministic
• Ek Erlang
• G General
• Example
• (M/M/1) (FCFS/inf/inf)

13
Single-Channel, Single-Phase(M/M/1) Formulas

• Probability of no units in the systemPo 1 - l
  / m
• Probability of n units in the systemPn (l /m)n
  (1 - l /m)
• Mean of units in systemL l / (m - l)
• Mean of units in the queueLq l 2 / m ( m - l
  )

14
(M/M/1) Formulas (Cont)

• Mean time in the systemW 1 / ( m - l )
• Mean time in the queueWq l / m ( m - l )
• Service Facility Utilizationp l / m
• Service Facility Idle TimeI 1 - l /m

15
FAX (or copier) Example

• Employees arrive at a rate of 20 per hour
• Assume a Poisson Distribution
• Average time at machine is 2 minutes
• Assume a Negative Exponential Dist.
• 1/ m
• One Machine with One Line(M/M/1) (FCFS/inf/inf)
  System

16

• Mean of units in systemL l / (m - l)L 20
  / (30 - 20)L 20 / 10L 2 employees in system

17

• Mean of units in the queueLq l 2 / m ( m - l
  )Lq 20 2 / 30 ( 30 - 20 )Lq 400 / 30 ( 10
  )Lq 400 / 300Lq 1.33 employees in line

18

• Mean time in the systemW 1 / ( m - l )W 1 /
  ( 30 - 20 )W 1 / 10 of an hour or W 6
  minutes each (on average) waiting for
  and using the machine

19

• Mean time in the queueWq l / m ( m - l )Wq
  20 / 30 ( 30 - 20 )Wq 20 / 30 ( 10 )Wq 20
  / 300Wq 1 / 15th of an hour orWq 4
  minutes each (on average) just
  waiting in line

20

• Service Facility Utilizationp l / mp 20 /
  30p .67 or 67 or the time
• Service Facility Idle TimeI 1 - l /mI 1 -
  20 / 30I 1 - .67I .33 or 33 of the
  time its idle

21
Should we hire an operator?

• An operator would reduce average service from 2
  minutes to 1.5 minutes
• The operator would cost 8.00 / hour
• Average employee wage is 10.20 hours
• L l / m - l 20 / (40 -20) 1 in system
• The average number of employees in system is
  reduced from 2 to 1

22
Resulting Costs

• Alternative Service Waiting Total
  Annual Cost Cost
  Cost Cost No Oper. 0
  20.40 20.40 40,800
  2 10.20 With Oper.
  8 10.20 18.20 36,400
  Annual Savings
  4,400