1. Tarbiat Modares University
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)
12/2/2018

2. What is a stochastic process?

3. 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

4. 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.)

5. Example: a Queuing System
Inter arrival times Y1, Y2, … (mutually independent) (FY)
Service times: S1, S2, … (mutually independent) (FS)
Notation for a queuing system: Fy /FY /m
Possible arrival/service time distributions types are:
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

6. 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
Discrete k
Discrete

7. 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)
Discrete
Continuous

8. 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
Continuous k
Discrete

9. 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

10. Further Classification
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
(1st order distribution)
(2nd order distribution)

11. 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.
Stationary: E[x(t)] = E[x]  ensemble average. When the pdf or the CDF exhibits stationarity property, then, the process is said to strictly stationary. If only the first moment satisfies this property, then, the process is said to stationary in the mean etc.

12. Markov Process

13. Markov Chains: Homogeneity

14. Markov Chains: Sojourn Time

15. 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.

16. Renewal Counting Process

17. Stationarity Process

18. 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)
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] =
{Yi} forms a discrete-time process.

19. Poisson Process A continuous time, discrete state process.
N(t): no. of events occurring in time (0, t]. Events may be,
# of packets arriving at a router port
# of incoming telephone calls at a switch
# of jobs arriving at file/computer server
Number of failed components in time interval
Events occurs successively and that intervals between these successive events are iid rvs, each following EXP( )
λ: average arrival rate (1/ λ: average time between arrivals)
λ: average failure rate (1/ λ: average time between failures)

20. Poisson Process (contd.)
N(t) forms a Poisson process provided:
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?

21. Merged Multiple Poisson Process Streams
Consider the system,
Proof: Using z-transform. Letting, α = λt, +

22. 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 +

23. 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 + X 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].

24. Renewal Counting Processes (contd.)
Sn t
For N(t), what is P(N(t) = n)?
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]. tn
More arrivals possible
F(n+1) (t): prob(time taken for n-renewals + time for one more renewal)
= tn + t

25. 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.

26. 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] =
Fn(t) will turn out to be
e–λ t (λ t)n/n! i.e Poisson process pmf
n-stage Erlang

27. 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 = E[Di+Ti+1] = E[Ti+Di]=E[Xi]=MTTF+MTTR
MTBF
T D T D T D3 T D4 …….

28. Availability Analysis (contd.)
Two mutually exclusive situations:
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
renewal
Repair is completed with in this interval x t

29. Availability Expression
dA(x) : Incremental availability
dA(x) = Prob(that after renewal, life time is > (t-x) & that the renewal occurs in the interval (x,x+dx])
Repair is completed within this interval x
x+dx t
Renewed life time >= (t-x)

30. 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,

31. Availability, MTTF and MTTR
Steady state availability A is:
for small values of s,

32. Availability Example
Assuming EXP( ) density fn for g(t) and w(t)