2. Simple Markovian

1
2. Simple Markovian Birth-Death Queueing Models
2.1 M/M/1 Queueing Model 2.2 Methods of Solving
Steady-State Difference Equations 2.3
M/M/c Queueing Model 2.4 M/M/c/K Queueing
Model 2.5 Erlangs Formula (M/M/c/c) 2.6 M/M/?
Queueing Model 2.7 Finite Source Queues 2.8
State-Dependent Service 2.9 Queues with
Impatience 2.10 Transient Behavior 2.11 Busy
Period Analysis
2
2.1 M/M/1 Queueing Model

• Arrival Process

• Service Process

• M/M/1 is a Birth-Death Markov Process

3
2.1 M/M/1 Queueing Model

• Rate Transition Diagram

• Stochastic Balance, Flow Conservation Law
• At a given state, the average total flow into
  the
• state equals the average total flow out of
  the state.

Global Balance Eqs.
4
2.1 M/M/1 Queueing Model

• Detailed (Local) Balance Equations

5
2.2 Methods of Solving S-S Difference Equations

• Iterative Method for M/M/1 Queue

6
2.2 Methods of Solving S-S Difference Equations

• Solving pn by Generating Functions

Global Balanced Eqs.
7
2.2 Methods of Solving S-S Difference Equations

• Solving pn by Generating Functions

8
2.2 Methods of Solving S-S Difference Equations

• Solving pn by Using Recurrence Equation

Global Balanced Eqs.
9
2.2.4 Measures of Effectiveness

• Mean System Size L

10
2.2.4 Measures of Effectiveness

• Mean Queue Size Lq

11
2.2.4 Measures of Effectiveness

• Mean Queue Size of Nonempty Queues

12
2.2.4 Measures of Effectiveness

• System Waiting Time and Queueing Delay

By Littles Formulas
13
2.2.4 Measures of Effectiveness

• Example
• Hair-Cut Shop,
• Poisson arrivals with rate of 5/hr,
• Customer processing time is exponential with
  
• mean of 10 min.

The average no. waiting when there is at least
one person waiting
The percent of time an arrival without waiting
If only 4 seats at present, Prfinding no seat?
14
2.2.4 Measures of Effectiveness

• Waiting Time Distributions

15
2.2.4 Measures of Effectiveness

• Waiting Time Distributions

16
2.2.4 Measures of Effectiveness

• System Time Distributions

Similarly, one can derive that
The system time has an exponential distribution
with parameter (???).
17
2.3 M/M/c Queueing Model

• Transition Diagram

• Solution for pn

Local Balanced Eq.
18
2.3 M/M/c Queueing Model

• Measures of Effectiveness

19
2.3 M/M/c Queueing Model

• Waiting Time Distributions

20
2.3 M/M/c Queueing Model

• Waiting Time Distributions

• System Time Distributions

21
2.3 M/M/c Queueing Model

• System Time Distributions

22
2.3 M/M/c Queueing Model

• Example
• Eye Clinic, free vision tests, 3 doctors
  on duty,
• Poisson arrivals with rate of 6/hr,
• Test time is exponential with mean of 20 min.

23
2.4 M/M/c/K Queueing Model

• Transition Diagram

Exercise consider the cases K ?, c 1.
24
2.4 M/M/c/K Queueing Model

• Measures of Effectiveness

How about ? 1 ?
25
2.4 M/M/c/K Queueing Model

• Measures of Effectiveness

For M/M/1/K
26
2.4 M/M/c/K Queueing Model

• Waiting Time Distributions

27
2.4 M/M/c/K Queueing Model

• Waiting Time Distributions

Note
28
2.4 M/M/c/K Queueing Model

• Example
• Automobile inspection station, 3 inspection
  stalls,
• each with room for only one car, the station
  can
• accommodate at most 4 cars waiting at one
  time,
• Poisson arrivals with rate of 1 car/min,
• Inspection time is exponential with mean of
  6 min.

29
2.5 Erlangs Formula M/M/c/c

• Transition Diagram

30
2.6 M/M/? Queueing Model

• Transition Diagram

• Example turning on TV sets Poisson with ?
  105/hr,

customers choose 5 TV stations at random,
viewing time Exponential with 1/? 1.5 hrs.
Ave. no. of viewers/station
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2.7 Finite Source Queues

• Transition Diagram for M Sources

32
2.7 Finite Source Queues

• Example
• A company has 5 robots, 2 repair people,
• When one is fixed, the time until the next
  breakdown is
• exponential with mean 30 hr,
• Repair time is exponential with mean of 3 hr.

33
2.7 Finite Source Queues

• Transition Diagram for M Sources and Y Spares

34
2.7 Finite Source Queues

• Transition Diagram for M Sources and Y Spares

35
2.7 Finite Source Queues

• Waiting Time Distribution

36
2.8 State-Dependent Service

• Markovian Queues with State-Dependent Service

37
2.8 State-Dependent Service

• Example
• Car-polishing machine has 2 speeds, 40 min
  on average at
• low speed, and 20 min on average at the high
  speed, to
• polish a car, the actual switching time is
  exponential.
• Poisson arrivals with mean interarrival time
  of 30 min,
• Switching Policies switching to high speed
  if there are any
• customer waiting, or switching to high speed
  only when
• more than one customer is waiting,

38
2.9 Queues with Impatience

• M/M/1 Balking

Possible examples for bn
If n people are in the system, an estimate for
the average waiting time might be n/? .
M/M/1/K is a special case of balking where bi 1
for 0 ? i ? K and 0 otherwise.
39
2.9 Queues with Impatience

• M/M/1 Reneging

Reneging function r(n)
A good possibility for the reneging function r(n)
is e?n/?, n?2.
40
2.10.1 Transient Behavior of M/M/1/1

• Differential Equations

41
2.10.2 Transient Behavior of M/M/1

• Differential Equations

42
2.10.2 Transient Behavior of M/M/1/?

• Solve the Inverse Laplace Transform

43
2.10.3 Transient Behavior of M/M/?

• Differential Equations

44
2.11 Busy Period Analysis

• Busy Period for M/M/1

Busy Period begin with the arrival to an idle
channel, and end when the channel next become
idle.
Idle Period if the arrival process is Poisson,
the idle period is exponential.
Transition Diagram for Busy Period no transition
from state 0 to state 1, and p1(0) 1.
45
2.11 Busy Period Analysis

• Busy Period for M/M/1

ETbp mean of busy period ETbc
mean of busy cycle
46
2.11 Busy Period Analysis

• Busy Period for M/M/c

i-channel busy period begin with an arrival at
the system at an instant when there are i-1 in
the system, and end at the very next point in
time when the system size dips to i-1. The system
busy period is the case where i 1.
47
End of Chapter 2