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In the Name of the Most High
Stochastic Processes
Behzad Akbari
Spring 2009
Tarbiat Modares University
These slides are based on the slides of Prof. K.S. Trivedi (Duke University)
2/18/2015
1
What is a stochastic process?
Stochastic Process Characterization

At a fixed time t=t1, we can define X(t1). Similarly, we can
define, X(t2), .., X(tk).
X(t1) can be characterized by its distribution function,

We can also consider the joint distribution function,

Discrete and continuous cases:



Index set T may be discrete/continuous
State space I (i.e. sample space S) may be discrete/continuous
Classification of Stochastic Processes



Four classes of stochastic processes:
Discrete-state process  chain
discrete-time process  stochastic sequence {Xn | n
є T} (e.g., probing a system every 10 ms.)
Example: a Queuing System
Queue (waiting station)
Random arrivals
Inter arrival time
distribution fn. FY
m
servers
Service time
distribution fn. FS

Inter arrival times Y1, Y2, … (mutually independent) (FY)

Service times: S1, S2, …

Notation for a queuing system: Fy /FY /m

Possible arrival/service time distributions types are:






(mutually independent) (FS)
M: Memory-less (i.e., EXP)
D: Deterministic
G: General distribution
Ek: k-stage Erlang etc.
M/M/1  Memory-less arrival/departure processes with 1-service station
Some examples: M/M/1, M/G/1, M/M/k, M/D/1
Discrete time, Discrete space Stochastic Processes

Nk: Number of jobs waiting in the system at the time of kth job’s
departure  Stochastic process {Nk|k=1,2,…}:
Nk
k
Discrete
Continuous Time, Discrete Space

X(t): Number of jobs in the system at time t. {X(t)|t є T} forms a
continuous-time, discrete-state stochastic process, with,
X(t)
Continuous
Discrete Time, Continuous Space

Wk: wait time for the kth job. Then {Wk| k є T} forms a Discrete-time,
Continuous-state stochastic process, where,
Wk
k
Discrete
Continuous Time, Continuous Space

Y(t): total service time for all jobs in the system at time t. Y(t) forms a
continuous-time, continuous-state stochastic process, Where,
Y(t)
t
Further Classification

(1st order distribution)





(2nd order distribution)
Similarly, we can define nth order distribution:
Difficult to compute nth order distribution.
Can the nth order distribution computations be simplified?
Yes. Under some simplifying assumptions:

Stationary (strict)

F(x;t) = F(x;t+τ)  all moments are time-invariant
Independent and Renewal Process

Independence

As consequence of independence, we can define
Renewal Process

Discrete time independent process {Xn|n=1,2,…} (X1, X2, ..
are iid, non-negative rvs), e.g., repair/replacement after a
failure.
Markov Process
Markov Chains: Homogeneity
Markov Chains: Sojourn Time
Markov Chain Sojourn time




Let, Y: time spent in a given state
Y is also called the sojourn time and has memoryless property:
This result says that for a homogeneous discrete time Markov chain,
sojourn time in a state follows EXP( ) distribution.
Semi-Markov process is one in which the sojourn time in state may not be
EXP( ) distributed.
Renewal Counting Process
Stationarity Process
Bernoulli & Binomial Processes


A set of Bernoulli sequences, {Yi|i=1,2,3,..}, Yi =1 or 0
{Yi} forms a Bernoulli Process. Often Yi’s are independent.


E[Yi] = p; E[Yi2 ] = p; Var[Yi] = p(1-p)
Define another stochastic process , {Sn|n=1,2,3,..}, where
Sn = Y1 + Y2 +…+ Yn (i.e. Sn :sequence of partial sums)




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Sn = Sn-1+ Yn
(recursive form)
P[Sn = k| Sn-1= k] = P[Yn = 0] = (1-p) and,
P[Sn = k| Sn-1= k-1] = P[Yn = 1] = p
{Sn |n=1,2,3,..}, forms a Binomial process
P[Sn = k] =
Poisson Process


A continuous time, discrete state process.
N(t): no. of events occurring in time (0, t]. Events may be,
3.
# of packets arriving at a router port
# of incoming telephone calls at a switch
# of jobs arriving at file/computer server
4.
Number of failed components in time interval
1.
2.

Events occurs successively and that intervals between
these successive events are iid rvs, each following EXP( )
1.
2.
λ: average arrival rate (1/ λ: average time between arrivals)
λ: average failure rate (1/ λ: average time between failures)
Poisson Process (contd.)

N(t) forms a Poisson process provided:
1.
2.
3.
N(0) = 0
Events within non-overlapping intervals are independent
In a very small interval h, only one event may occur (prob. p(h))

Letting, pn(t) = P[N(t)=n],

Hence, for a Poisson process, interval arrival times
follow EXP( ) (memory-less) distribution. Such a
Poisson process is non-stationary.
Mean = Var = λt ; What about E[N(t)/t], as t  infinity?

Merged Multiple Poisson Process Streams

Consider the system,
+

Proof: Using z-transform. Letting, α = λt,
Decomposing a Poisson Process Stream

Decompose a Poisson process into multiple streams
+




N arrivals decomposed into {n1, n2, .., nk}; N= n1+n2, ..,+nk
Cond. pmf
Since,
The uncond. pmf
Renewal Counting Process

Poisson process  EXP( ) distributed inter-arrival times.
What if the EXP( ) assumption is removed  renewal proc.
Renewal proc. : {Xi|i=1,2,…} (Xi’s are iid non-EXP rvs)
Xi : time gap between the occurrence of ith and (i+1)st event

Sk = X1 + X2 + .. + Xk  time to occurrence of the kth event.




N(t)- Renewal counting process is a discrete-state,
continuous-time stochastic. N(t) denotes no. of renewals in
the interval (0, t].
Renewal Counting Processes (contd.)
Sn

t
For N(t), what is P(N(t) = n)?
tn
More arrivals possible

1.
2.
nth renewal takes place at time t (account for the equality)
If the nth renewal occurs at time tn < t, then one or more renewals
occur in the interval (tn < t].
Renewal Counting Process Expectation

Let, m(t) = E[N(t)]. Then, m(t) = mean no. of arrivals in
time (0,t]. m(t) is called the renewal function.
Renewal Density Function

Renewal density function:

For example, if the renewal interval X is EXP(λ x), then



d(t) = λ , t >= 0 and m(t) = λ t , t >= 0.
P[N(t)=n] = e–λ t (λ t)n/n! i.e Poisson process pmf
Fn(t) will turn out to be n-stage Erlang
Availability Analysis



Availability: is defined is the ability of a system to provide
the desired service.
If no repairs/replacements, Availability = Reliability.
If repairs are possible, then above def. is pessimistic.
MTBF
T1

D1
T2
D2
T3
D3
T4
D4 …….
MTBF = E[Di+Ti+1] = E[Ti+Di]=E[Xi]=MTTF+MTTR
Availability Analysis (contd.)
renewal
x

t
Two mutually exclusive situations:
1.
2.

Repair is completed with in this interval
System does not fail before time t  A(t) = R(t)
System fails, but the repair is completed before time t
Therefore, A(t) = sum of these two probabilities
Availability Expression
dA(x) : Incremental availability

Repair is completed within this interval
0
x
x+dx
t
Renewed life time >= (t-x)

dA(x) = Prob(that after renewal, life time is > (t-x) & that
the renewal occurs in the interval (x,x+dx])
Availability Expression (contd.)

A(t) can also be expressed in the Laplace domain.

Since, R(t) = 1-W(t) or LR(s) = 1/s – LW(s) = 1/s –Lw(s)/s

What happens when t becomes very large?

However,
Availability, MTTF and MTTR

Steady state availability A is:

for small values of s,
Availability Example

Assuming EXP( ) density fn for g(t) and w(t)
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