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Chapter Chapter 11
IntroductionIntroduction
CONTENTS
1.1 DESCRIPTION OF THE QUEUEING PROBLEMS..............................................- 4 -
1.2 CHARACTERISTICS OF QUEUEING PROCESSES............................................- 5 -
1.3 NOTATION.......................................................................................................- 9 -
1.4 MEASURING SYSTEM PERFORMANCE........................................................- 10 -
1.5 SOME GENERAL RESULTS...........................................................................- 11 -
1.5.1 LITTLE’S FORMULAS........................................................................- 13 -
1.6 SIMPLE DATA BOOK KEEPING FOR QUEUES.............................................- 16 -
1.7 POISSON PROCESS AND THE EXPONENTIAL DISTRIBUTION.....................- 21 -
1.8 MARKOVIAN PROPERTY OF THE EXPONENTIAL DISTRIBUTION..............- 27 -
1.9 STOCHASTIC PROCESSES AND MARKOV CHAINS......................................- 29 -
1.9.1 MARKOV PROCESS............................................................................- 30 -
1.9.2 DISCRETE-PARAMETER MARKOV CHAINS......................................- 32 -
1.9.3 CONTINUOUS-PARAMETER MARKOV CHAINS.................................- 34 -
1.9.4 IMBEDDED MARKOV CHAINS...........................................................- 39 -
1.9.5 LONG-RUN BEHAVIOR OF MARKOV PROCESS: LIMITING
DISTRIBUTION, STATIONARY DISTRIBUTION, ERGODICITY............- 41 -
1.9.6 ERGODICITY......................................................................................- 45 -
1.10 STEADY-STATE BIRTH-DEATH PROCESSES.............................................- 50 -
2
Queueing System (排隊系統)
Wait in line at post office, supermarket, highway, …, etc.
Wait in line within a time-sharing computer system.
Wait in line before a server-farm system.
Wait in line before a switch/router system.
Wait in line before a protocol layer program module.
Wait in line in a statistical multiplexer.
Design Issues:
Design System Parameters so that the queueing system has
an optimal performance
Channel allocation in GSM/WCDMA/OFDMA systems.
Buffer design in switch/ router communication systems…
Bandwidth allocation for statistical multiplexer,
e.g. EPON system.
Scheduling in the server-farm or switch/router system.
3
Performance Measures
How long must a customer wait?
→ waiting time and system time
How many people will form in the line?
→ average queue length
How is the productivity of the server (counter)?
→ throughput, utilization
How much is the blocking probability? Or say how many
communication systems should be prepared?
“Queuing Theory” attempts to answer these questions through
detailed mathematical analysis.
4
1.1 DESCRIPTION OF THE QUEUEING PROBLEMS
Fig. - Conceptual Queuing Model
Arriving customer could be connection request asked for
connection setup and minimum data rate service.
The queueing model is commonly used in the traffic control,
scheduling, and system design.
5
1.2 CHARACTERISTICS OF QUEUEING PROCESSES
Arrival Pattern of Customers
The arrival pattern is measured in terms of probability
distribution or mean of arrival rate or interarrival time.
The arrival pattern may be one customer at a time or bulk.
In bulk-arrival situation, both the interarrival time and the
number of customers in the batch are probabilistic.
Customer reactions:
patient customers
balked (blocked) customer
reneged customer
jockey customer (switch from one line to another)
Stationary Process (time independent) vs. Non-stationary
Process (Not time independent, correlated)
Service Pattern
Service time distribution (random process)
service may be single or batch
state-independent service or state-dependent service
Impatient ustomers
Service timeService rate
6
Queue Discipline (Service Discipline)
First In, First Out (FIFO)
Last In, First Out (LIFO)
Selection for Service in random order (SIRO), independent
of time arrival to the queue, Most delay first serve (MDFS)
Priority
1. Preemptive
2. Non preemptive
The Round-Robin Service Discipline
(Operating System Theory, E.G. Coffman, Jr.)
7
The shortest-elapsed-time Discipline
System Capacity
In some queueing process there is a physical limitation to
the amount of waiting room (buffer size), i.e., system
capacity is finite
Fig. 1.2 - Multichannel queueing system
8
Number of Service Channels
Multiple channel queueing system
Routing and buffer sharing
Stages of Service
Fig. 1.3 - A multistage queueing system with feedback
Consider a communication network with ARQ recovery
control. It is a kind of multi-stage queueing system.
“Before performing any mathematical analysis, it is absolutely
necessary to describe adequately the process being modeled.
Knowledge of the aforementioned six characteristics is “Essential”
in the queueing analysis.”
9
1.3 NOTATION
Table 1.1 - Queueing Notation A/B/X/Y/Z
10
1.4 MEASURING SYSTEM PERFORMANCE
Blocking probability (Grade of service)
Waiting time or mean system time
(Quality-of-Service, requirement)
Mean queue length
(Average number of customers in the system)
System utilization (Throughput)
11
1.5 SOME GENERAL RESULTS
Consider the queuing model (G/G/1 or G/G/C)
Fig. - G/G/C Queueing Model
If , the queue will get bigger and bigger, the
queue never settles down, and there is no steady state.
If , unless the processes of arrivals and
service are deterministic and perfectly scheduled, no steady
state exists, since randomness will prevent the queue from ever
empty out and allowing the serves to catch up, time causing the
queue to grow without bound.
12
Notation Description
The number of customer in the system at time t
The number of customer in the queue at time t
The number of customer in the service at time t
System time
Waiting time
Service time
Consider C-Server queue
At steady state :
13
14
1.5.1 LITTLE’S FORMULAS
John D.C. Little related to ,
John D.C. Little related to ,
where , , and .
Little’s Formulas :
(1.1a)
(1.1b)
Fig. 1.4 - Busy-period sample path
15
16
A concept proof
(1.2a)
(1.2b)
(1.1a)
(1.1b)
Any queueing systems:
17
(1.3)
The expected number of customers in service in steady state.
For a single-server system :
For multiple-server system
For multiple server system,
Table 1.2 - Summary of General Result for G/G/C Queue
18
Little’s formula
Little’s formula
Expected-value argument
Busy probability of an arbitrary server
Expected number of customers in service ; Offered workload
Traffic intensity; workload to a server
※
19
1.6 SIMPLE DATA BOOK KEEPING FOR QUEUES
From fig. 1.5, we can see that
Fig. 1.5 – Sample path for queueing process
Thus using input data shown in Table 1.3
Table 1.3 – Input Data
20
We can have event-oriented book keeping shown on Table 1.4
Table 1.4 – Event-Oriented Bookkeeping
21
Column (2) : Arrival/Departure Customer i
Master Clock Time Arrival/DepartureCustomer i
0 1-A1 1-D2 2-A3 3-A5 2-D
Column (3) : Time of arrival i enters service
Column (4) : Time of arrival i leaves service22
Column (5) : Time in Queue:
Column (3) - Column (1)
Set
Column (6) : Time in System:
Column (4) - Column (1)
Column (7) : No. in Queue just after Master Clock Time:
A’s in Column (2) - D’s in Column (2) - 1
Column (8) : No. in System just after Master Clock Time:
A’s in Column (2) - D’s in Column (2)
23
Check Little’s Formula
Average time(by Column (6)) 70/12 = W
Average Queue Length (by Column (6))
70/31
Mean arrival rate
Average length
24
1.7 POISSON PROCESS AND THE EXPONENTIAL DISTRIBUTION
The most common stochastic queuing models assume that:
The arrival rate and service rate follow a Poisson distribution,
or equivalently, the interarrival times and service times obey the
exponential distribution.
Consider an arrival process , where denotes
the total number of arrivals up to t, with , and which
satisfies the following these assumptions:
1.
where is the arrival rate, and
2.
3. The number of arrivals in non overlapping intervals is
statistically independent.
tt t
t
25
Then, we have
(1.6)
(1.7)
We have
For the case , we have
26
where
27
Divide the above two equations by and as , we have
(1.11)
(1.12)
Then we have
We conjecture the general formula to be
(1.14)
28
That is a Poisson distribution.
This can be proven by mathematical induction.
29
We now show that if the arrival process follows the Poisson
distribution, the interarrival time follows the exponential
distribution.
Proof
Let T be the Random variable of “time between two arrivals”,
then
Let be the CDF of T,
Thus T has the exponential distribution with mean arrival time .
30
31
On the contrary, it can be shown that if the interarrival times
are independent and have the same
exponential distribution, then the arrival process follows the
Poisson distribution.
Proof
Let denote : CDF
Then
(1.15)
where is an Erlang distribution which is the sum
of n+1 independent and identical exponential random
variables.
Also
32
Let
Via manipulation, we have
That is a Poisson process.
※ The arrival time is uniformly distributed over the time axis.
33
1.8 MARKOVIAN PROPERTY OF THE EXPONENTIAL DISTRIBUTION
Proof
The exponential distribution is the only continuous
distribution which exhibits this memoryless property.
The proof of the above assertion rests on the fact that the
only continuous function solution of the equation
34
is the linear form
(1.18)
Proof of if , then
Proof
The memoryless property (1.17) can be rewritten as CCDF
(1.19)
Let
Take natural logarithm
Batch Poisson
35
Use the above-mentioned fact, we have
There are many possible and well-known general equations of
the Poisson/exponential process and will be taken up in great
detail in the text
36
1.9 STOCHASTIC PROCESSES AND MARKOV CHAINS
A stochastic process : , a family of random variables.
is defined over some index set or parameter space T.
T : Time range,
: state of the process at time t.
If T is a countable sequence, for example,
Then is said to be a discrete-parameter process
defined over the index set T.
If T is an interval, for example,
Then is called a continuous-parameter process
defined over the index set T.
Stationary:
37
Wide-sense stationary (W.S.S.):
independent of t
is a function of
38
1.9.1 MARKOV PROCESS
Continuous-parameter stochastic process: or
Discrete-parameter stochastic process:
is a Markov process
if for any set of n time points
in the index set or time range of the process, the conditional
distribution of , depends only on the immediately
preceding value . More precisely,
“Memoryless”: given the present, the future is independent of the
past.
Classification of Markov process:
According to:
(1) the nature of the parameter (index set, time range) space
of the process
(2) the nature of the state space of the process
39
Table 1.5 – Classification of Markov Processes
Parameter space (T)
State space Discrete Continuous
DiscreteDiscrete parameter
Markov chain
Continuous parameter
Markov chain
ContinuousDiscrete parameter
Markov process
Continuous parameter
Markov process
Semi-Markov Process (SMP) or Markov Renewal Process
(MRP)
If the time between two consecutive transitions T is an
arbitrary random variable, the process is called SMP.
If T is exponentially distributed for continuous parameter
cases, or geometrically distributed for discrete parameter
cases, then the SMP is reduced to Markov process.
40
1.9.2 DISCRETE-PARAMETER MARKOV CHAINS
The conditional probability
: Transition probability (single-step)
If these probabilities are independent of n, then the Markov chain
is called homogeneous chain.
: Transition Matrix
For homogenous chain, the m-step transition probability
are also independent of n.
From basic laws of probability,
In matrix notation
if ,
and then , where
Define the unconditional probability of state j at the mth trial by
: state probability at mth trial.
The initial distribution is given by .
Chapman-Kolmogorov (C-K) equations ustomers
41
, where
(1.24)
( →The sum of all rows equal to 0)
If limiting probability exists, at steady
state (Q→The sum of all rows equals to zero) or
42
43
1.9.3 CONTINUOUS-PARAMETER MARKOV CHAINS
, for and is countable for
.
From C-K equations, intuitively,
(1.25)
In matrix notation
Letting ,
44
: state probability of j at time t, regardless of starting state.
※ 讀課本 Poisson process的例子 p.30
45
Additional Theory:
If (1)
The prob. of change is linearly propositional to , with
propositionality constant which is a function of i and t.
(2)
Then we have Kolmogorov’s forward and backward equations;
respectively, by
(1.28)
46
Let and assume a homogeneous process as that
, for all t
Multiplying by
In matrix notation
, where and
At steady state, , where
Note that . ← It is intuitive
47
Since
we can have
and since
Q: Intensity matrix, where
For Poisson process (pure birth process)
48
與(1.12)相同
For birth-death process
The Poisson process is often called a pure birth process.
49
50
1.9.4 IMBEDDED MARKOV CHAINS
If we observe the system only at certain selected times, and the
process behaves like an ordinary Markov Chain, we say we have
an imbedded Markov Chain at those instants. (turn our attention
away from the truly continuous-parameter queuing process to an
imbedded discrete-parameter Markov Chain queuing process)
Consider the birth-death process at transition time
※
51
The transition probability ,
52
1.9.5 LONG-RUN BEHAVIOR OF MARKOV PROCESS: LIMITING DISTRIBUTION, STATIONARY DISTRIBUTION, ERGODICITY
If , for all i (independent of i),
We call the limiting probability of the Markov Chain.
(Steady-state probability)
Consider the unconditional state probability after m steps,
or equivalently , together with boundary condition
. These well-known equations are called stationary
equations, and their solution is called stationary distribution of the
53
Markov Chain.
If the limiting distribution exists, this implies that the resulting
stationary distribution exists and implies that the process
possesses steady-state. (The stationary distribution = the limiting
distribution)
But the converse is not true.
Example 1.1
The stationary distribution of Eq. (1.32), , is existed
(a solution to is existed), but there is no , and
does not exist except
(i) limiting 不存在(ii) stationary distribution不存在(iii) → strictly stationary
(iv) the process still does not possess steady-state
Strictly stationary: for all k and h
54
i.e. possesses time-independent distribution functions.
is independent of m.
The solution to does not imply strict stationary, except
But strict stationary does imply that is time-
independent, not independent of time.
Example 1.2
The process possesses (i) a steady-state since ,
(iii) but not in general stationary unless , and (ii) It
is strictly stationary at .
Example 1.3
55
and we have the stationary solution .
(i) limiting distribution exists
(ii) stationary distribution exists
(iii) the process is not completely stationary unless
only in the limit.
For continuous-parameter processes, the stationary solution can be
obtained from .
Thus, if the limiting distribution is known to exist, the solution
can be obtained from
56
57
1.9.6 ERGODICITY
is ergodic if time averages (statistics) equal ensemble
averages (statistics), where
Time-average:
Continuous-parameter Discrete-parameter
Ensemble-average:
58
59
※
=…
Ergodic → All moments are equal (The same statistics)
Example 1.1
The process is not ergodic if
The process is ergodic if (But stationary)
Example 1.2
The process is ergodic (not stationary), but two states, i and j,
are said to Communicate (i↔j)
If i is accessible from j (j→i) and j is accessible from i (i→j)
If all of its states communicate, the Chain is called
irreducible Markov Chain.
60
The state is aperiodic if
The Chain is said to be aperiodic if all states are aperiodic.
: The probability that the chain ever returns to j
If
when , define
Note: The stationary process is ergodic, but the ergodic process
need not be stationary.
Example in p.51
※ state 1 is recurrent,61
※ states 0,3,4 are recurrent,
※ states 2,5 are transient.
62
Theorem 1.1
(a) an irreducible, positive recurrent discrete-parameter
Markov Chain
(b) →The process becomes stationary→Ergodic
(c) If the Markov Chain is irreducible, positive recurrent, and
aperiodic, then the process is ergodic, and has a limiting
prob. distribution = stationary prob. distribution.
Theorem 1.2
An irreducible, aperiodic chain is positive recurrent, if there
exists a nonnegative solution of the system.
such that
63
Theorem 1.3
For continuous-parameter Markov Chain, the imbedded
Markov Chain need not be aperiodic as long as the holding
times in all states are bounded for Theorem 1.1 to be valid.
64
65
1.10 STEADY-STATE BIRTH-DEATH PROCESSES
At steady state,
From
we have
can be proven by induction.
66
Since ,
67
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