Traffic Analysis and Design of Wireless IP Networks phần 4 docx

Due to the memoryless property of the Markov chains, we may write the following: P{system remains in state i after m consecutive steps} = 1 – pii p ii m 4.27For continuous-time Markov ch

In discrete-time Markov chains, the time that the system spends in thesame state is geometrically distributed [2] We can easily prove this statement.

Let us assume that the system has entered a state i Then, the probability that the system will remain in the same state is p ii The probability that the system will

leave its state at the next step is (1 – pii) Due to the memoryless property of the

Markov chains, we may write the following:

P{system remains in state i after m consecutive steps} = (1 – pii )p ii m

(4.27)For continuous-time Markov chain, we have exponential distribution

of the time in single state (discrete-state continuous-time Markov process, seeFigure 4.3), and we may write the following:

The probability that the interarrival time between two consecutive arrivals

will be up to t after it was t0may be calculated by

If there is only one event from t = 0 to time t = t0, then the probability for

a new event to occur in next time period t (from t0to t + t0) does not depend

upon t0

We will further apply Markov processes in telecommunications becausemost of the random events can be considered in a Markov chain fashion

4.4 The Birth-Death Process

The birth-death process is a special case of the Markov processes Here, the sitions are permitted between adjacent states only We are mainly interested incontinuous-time processes, so we consider birth-death processes in that fashion.The probability that more then one event will occur in an infinitesimal timeinterval is zero:

pri-some state E kaccording to the number of users So, without losing generality, we

may denote with E k the state of the system when the population is of size k From the state k, birth-death process may transit only in state k + 1 and state k –

1, or remain in the state k during time interval ∆t We introduce the notion of

birth rateλkas well as death rateµk in a state k Due to the memoryless property

of the birth-death process, these birth and death rates are independent of time,

but depend upon the current state E konly Possible transitions in a birth-deathprocess are shown in Figure 4.5

Because the birth-death process permits transitions among neighboringstates only, we can describe it by a state transition diagram shown in Figure 4.6

We will refer to this type of diagram as a one-dimensional Markov chain

The state Ek can be reached in time interval ∆t from states Ek–1, Ek, and

E k+1 Considering that birth and death are independent and using Figure 4.5, wemay write:

1 The probability of exactly one birth in (t, t + ∆t) when the process is

Let us denote with p ij the probability for a transition from state i to state j.

Using the Kolgomorov-Chapman approach, we analyze possible transitions of

our particle In this case, in a time interval (t, t + ∆t) we can enter state Ekonly

by three mutually exclusive possibilities:

1 P{no state change occurred in state k}= [1 –λk∆t + o(∆t)] [1 –µk∆t + o(∆t)];

2 P{the system was in state k – 1 and we had one birth} = λk–1∆t +

P t0 + µ1 1P t ,k =0

(4.36)

We have lower boundary at the state 0 (no population) Also, it is possible

to have an upper boundary if we have specified the maximum number of users

in the system Because birth-death process is a special case of the Markovprocesses, we can apply the same matrix notation introduced by (4.11) to(4.13), and we obtain the following transition matrix:

Let us now assume, for simplicity, that the system starts at state E0at time

The last relation is called Poisson distribution It characterizes the Poissonprocess, which, in fact, is a pure birth process with

0

The Poisson process is significant in traffic theory in telecommunications,especially for circuit-switched networks, as we shall see later in this chapter Butthe Poisson process has an even wider significance It was shown by Palm that inmany cases a large sum of independent stationary renewal processes tends to aPoisson process

4.4.1 Stationary System

In practice we are interested in a stationary regime of processes because it is venient for the analysis due to unique distribution of the state probabilities,independent of the initial condition For a stationary system it holds that

dP t dt

system, p i = 0 for i < 0, andλi = 0, i < 0;µj = 0, j < 1 Because the

birth-death process in statistical equilibrium is a case of Markov processes, we canapply the general equation for a Markov chain in equilibrium, which can bederived directly from the state-diagram given in Figure 4.7

To obtain dependences among state probabilities, we draw arbitraryboundaries as shown in Figure 4.7 The total outgoing rate from a closedboundary should be equal to the total incoming rate into the boundary in a state

of equilibrium Using boundary 1 from Figure 4.7, we obtain

104 Traffic Analysis and Design of Wireless IP Networks

TE AM

FL Y

µ1 1P =λ0P0 (4.46)

and so on; from ith boundary we get

λi−1P i−1 =µi P i (4.47)Also, we require the conservation relation to hold for state probabilities:

0

1 0 1 1

11

=+

1

1 0 1 1

2

0 1

1 1

All states will be ergodic only and only if S1<∞ and S2=∞ Because weneed an ergodic process to have equilibrium, it is of most interest to our analysis.

The condition is fulfilled if there exists some k0such that for all k > k0it holdsthat

4.4.2 Birth-Death Queuing Systems in Equilibrium

Let us consider the importance of the statistical equilibrium of a birth-deathprocess We can define two types of equilibrium: a global balance and a localbalance

Global balance may be defined by using (4.44) and by applying it in

infini-tesimal time interval∆t:

λk∆tP k + µk∆tP k = λk−1∆tP k−1 + µk+1∆tP k+1 k≥1 (4.54)

By analyzing the last equation, we may observe that the left side of the tion gives the probability of a transition to neighboring states with respect to

rela-k—that is, to k + 1 (a new birth), and to k – 1 (a death) The right side of (4.54)

gives the probabilities of transition from adjacent states to the state k We can say that total outgoing traffic intensity from a particular state k is equal to the

total incoming traffic intensity to that state This is referred to as a globalbalance

Local balance is defined by multiplying (4.47) by ∆t, which leads to

λi−1∆tP i−1 = µi∆tP i fori =1 2 3, , , (4.55)From the last equation it is obvious that the possibility of a transition from

state k – 1 to state k (the left side) is equal to the transition probability in the reverse direction (the right side of the equation) This is called a local balance, and (4.55) is referred to as a local balance relation.

4.5 Teletraffic Theory for Loss Systems with Full Accessibility

We covered the basics of queuing theory in previous sections of this ter Now, let us go through traditional way of design and analysis in telecommu-nications represented by the famous Erlang’s loss formula (elsewhere it isreferred to as Erlang-B formula or Erlang’s first formula)

chap-In the early decades of telephony and switching systems, Erlang made anextensive analysis of the traffic data such as telephone calls initiated by usersconnected to a switching system (telephone exchange), blocking of the calls andtheir duration He found that call arrivals suit well into a Poisson process Also,call duration was shown to be easily modeled by using the exponential distribu-tion for the call duration times According to the above statement, we may saythat Erlang’s loss formula is based on the following model:

• Arrival process is Poisson and service times are exponentially distributed

• We consider a circuit-switched system with servers (channels, trunks, ortime slots) working in parallel

• An arrival is accepted for service if any channel is idle The system cates one channel per call We say the group (of channels) has full acces-sibility when every incoming user competes with other users for all idlechannels (not allocated resources)

allo-Using the queuing theory and Kendall notation for queuing systems [2],

we can describe Erlang’s conclusions by using M/M/n/n queuing system In this case n servers are n channels that may serve up to n users at the same time Usu-

ally, the number of potential users is many times higher than the number ofavailable channels (resources) due to the economic aspects of telecommunica-tions networks design This statement holds for both analog and digital circuit-switched networks, because in both cases we have one type of traffic only (voicetelephony) and the system allocates equal resources for each call Of course,because telephony is bidirectional, we have occupancy of two channels per call(one for each direction), but in traditional traffic theory it is enough to consideronly one direction in calculations due to symmetrical resource allocation in bothdirections This picture will change, however, if we introduce packet-basedcommunication and heterogeneous services, as we shall later see

The state diagram for M/M/n/n is given in Figure 4.8 For this system we

have

k k

(4.56)

λµ

Using (4.49) and (4.50), which we proved for the general case of thebirth-death processes, and by replacingλkandµkaccording (4.56), we obtain

i

A k A i

k

k

i

i n

k i i n

λµ

where A =λ/µis intensity of the offered traffic It is expressed in units Erlangs in

honor of Erlang Relation (4.57) is called Erlang distribution or truncated

Pois-son distribution.

Definition of the carried traffic: We define the carried traffic per single

channel as a sum of busy times for each channel during time interval T divided

by the time interval For a pool of resources, it is given by

( )

A

x t dt T

depend upon the moment in time—that is, P(x, t) = P(x) Then, the offered

traffic may be calculated by using the following equation:

Also, we may define offered traffic as average intensity of calls C A

(calls/sec-ond) multiplied by average call duration time tµ:

It is obvious that C A=λ, while tµ= 1/µdue to exponential distribution ofthe call duration, so we get

n i i n

The last relation is called Erlang’s loss formula, written for the first time in

1917 by Erlang It is also denoted as E 1,n (A) = En (A) where index 1 indicates

that it is Erlang’s first formula (Erlang-B formula)

For the sake of completeness, we briefly refer to two other situations

regarding the number of users N and the number of trunks n—that is, N ≤n and

N >n (not N>>n) In the Erlang’s case, the number of users N>>n and hence number of idle users is N – n >>n However, for these two cases, we do not

have independence of the offered traffic from the number of busy trunks fore, for both of them the call arrival rate depends upon the number of activeusers [i.e.,λk = (N – k)λ,µk = kµ], for k busy trunks In these cases, we use an

There-additional parameter β=λ/µ, which is offered traffic per idle source If we sider that the source changes between idle and busy states, the offered traffic persource is

ββ

1

/

Then, the offered traffic to the system is A = Nα Using the state

transi-tion diagram given in Figure 4.8, we can derive the distributransi-tion functransi-tions

Hence, for the case N≤n we get the Binomial distribution:

0

(4.66)

The Erlang-B formula has been used throughout the last century, and itcontinuous to be used today for design and analysis of circuit-switched tele-phone networks with wired (fixed) access With some approximations, it may beused in mobile cellular networks We will refer to wireless networks later in thischapter.

When we design a system we need to specify some grade of service (GoS)

or QoS requirements (we define these later in this chapter) For that purpose,

we define performance parameters, which indicate the QoS (or GoS) level

of the designed system The main QoS parameter in circuit-switched networks

is blocking We define three types of blocking: call congestion, time congestion,and traffic congestion In the following we consider the Erlang distribution.Call congestion is the probability that a random call is lost due to block-ing; that is, all channels are busy at call arrival:

λµ

When number of users is many times greater then the number of channels(assumption holds for Erlang’s loss formula), we have

This is a characteristic of all systems with the Poisson arrival process and alarge population In Figure 4.9 we show blocking as a function of the offeredtraffic

We have three related parameters in Erlang’s first formula: number of

channels n, offered traffic A, and blocking E 1,n (A) If we know two of them, we

can calculate the third parameter For example, if we design a switching system,

we need to predict the offered traffic to the system and specify the desired GoS.Then, we can obtain the number of needed channels by applying Erlang’s firstformula Although we derived the formula starting from the exponential distri-bution of connection holding times, it can be easily shown that it is independentfrom the holding time distributions The basic assumption in Erlang’s loss for-mula is Poisson arrival process, which is fulfilled only when we have a largenumber of sources, according to Palm’s theorem This assumption is valid intelephone systems and that is why the formula has been widely used since thesecond decade of twentieth century

4.6 Teletraffic Theory for Loss Systems with Multiple Traffic Types

Today, in telecommunications networks we usually have more than one traffictype (the voice telephony) Different traffic types have different characteris-tics considering call arrivals, intensities, and call duration times Networks with

integrated heterogeneous traffic sources are called integrated networks (it is anotation for a circuit-switched networks) Each service has different traffic char-acteristics In networks with asynchronous transport of the information, differ-ent traffic services have different intensities Typical examples of asynchronoustransmission are packet-based networks.

4.6.1 Loss Systems with Integrated Traffic

In packet-based networks as well as in integrated networks, we often have vals with different rates and different resource demands from different traffictypes If we assume that a channel (or we may say a bandwidth allocation unit) isthe smallest unit that can be allocated by the system, then we can have allocation

arri-of two, three, or more bandwidths units (e.g., channels) for some connections(e.g., video streaming)

We analyze the teletraffic system M[ξ]/M/n/k This is full accessibility

group of resources with possibility of losses The distribution of the connectionsinterarrivals times and connection holding times are assumed to be exponential.Theξis a random variable, which shows the occupied resources as a function of

time Let b i denote the probability that i bandwidth units are allocated to a call:

We will refer again to resources as channels So, from every state i= 0, 1,

, n – 1 we have arrival rate , but varying number of requested channels per call

(i.e., 1 channels/call, 2 channels/call, and so on) Using the condition for aglobal balance, from Figure 4.10 we may write

Of course, the conservation of state probabilities must hold:

By summing the equations (4.75) from 0 to j – 1 we obtain a recurrent

relation for the state probabilities:

P

k j i

Probability distribution forξmay vary For an example,ξmay be

geomet-rically distributed [4] Then, one can calculate the probability that i-channels

are simultaneously occupied by using

where p is the probability that a call requests one channel When using rical distribution, the average number of busy channels allocated per call is b=

geomet-1/p After some algebra, for the state probabilities we obtain

4.6.2.1 Steep and Flat DistributionsSometimes, if we cannot describe a random process (e.g., call duration) with sin-gle exponential distribution (i.e., one parameter), then we can use two or moreexponential distributions There are two basic types of such combinations ofexponential distributions (introduced by Palm):

1 Steep distribution, which is a set of stochastic independent exponential

distributions in series (Figure 4.11);

2 Flat distribution, which is a set of stochastic independent exponential

distributions in parallel (Figure 4.12)

114 Traffic Analysis and Design of Wireless IP Networks

We refer to steep distribution as generalized Erlang distribution It is

obtained by convolving k exponential distributions If all k exponential tions are identical, we get an Erlang-k distribution (we refer only to this case)

distribu-with the following probability distribution function:

have parallel combination k exponential distributions with parameters,λ1,λ2, ,

λk , and discrete positive weights W(λ) (i.e., p1, p2, , p k) where

4.6.2.2 Coxian Distributions

Coxian distribution is a general distribution obtained by combining steep andflat distributions Each exponential distribution that is used in the Coxian distri-bution (or in other specific cases, such as steep or flat distributions) is called anexponential phase Therefore, we refer to Coxian distribution as a general class

of phase-type distributions Elsewhere it is called Erlang distribution withbranches

The Coxian distribution function can be written as a weighted sum ofexponential distributions But, different from flat distributions where the sum ofall weights equals one, as given by (4.85), and all weights are positive, in Coxiandistribution we have the following conditions for weights:

X2, or X = X1+ X2+ + Xk So, a random variable has a Coxian

distribu-tion of order k if it has to go through up to at most k exponential phases The mean length of phase n isλn , n = 1, 2, , k It starts in phase 1 At each stage n,

p n is probability that a job leaves the server immediately after completing

stage n, and q n = 1 – pn is the probability that a job requires more service

after stage n, as shown in Figure 4.13.

Coxian distribution gets a lot of attention due to its wide applicability todistributions of practical interest and also because of the possibility of using thetheory of Markov processes (phase-method), which do not require advanced

mathematics For example, Erlang-k distribution and hyper-exponential

distri-bution are special types of Coxian distridistri-bution We may also show exponentialdistribution through its Coxian distribution equivalent

4.6.3 Multidimensional Erlang Formula

We generalize the teletraffic theory to systems with multiple traffic types Insuch systems we have different classes of services (i.e., services with various traf-fic and QoS demands) Each class corresponds to a traffic stream We willexpand the Erlang loss formula to a general case of a network with multiple traf-fic classes

For that purpose, we consider a group of n bandwidth units (e.g.,

chan-nels, time slots), which is offered to two independent traffic streams with val/service rates (λ1,µ1) and (λ2,µ2 ) Then, we have the offered traffic A1= λ1/µ 1

arri-for the stream 1, and A2=λ2/µ2for the stream 2

Let us denote with (i, j) the state of the system when there are i tions from stream 1 and j connections from stream 2 We limit allocation to one

connec-channels or slot per connection Then, the following restrictions hold: 0≤i≤n,0≤j ≤n, and 0≤i + j≤n In this case we have a two-dimensional transition-state diagram, given in Figure 4.14, that corresponds to a reversible Markovprocess

By assuming equilibrium in the system, we may apply global balance

equa-tions If we denote with P(i, j) the probability that the system is in state (i, j),

then we may write

i

A j

A k