CS457 - System Performance Evaluation - Winter 2008

Questions and Comments

Tutorial: MC4061, Monday March 17, 18.00 to 19.00

Lecture 28

Analytic Queueing Theory

Text: Chapters 30--36.

Markov process

A stochastic process in which S(n+1) is independent of S(0), ..., S(n-1), but may depend on S(n).

In performance evaluation, when we talk of a Markov process we usually also mean that the next transition occurs at a time distributed by an exponential distribution.

In a Markov process with exponentially distributed transitions the mean transition rate is the only parameter that can vary with state.
That is, if in state S(j) then the transition rate is \lambda_j

Another useful property of the exponential distribution

Y1 - arrival time - exponential - \lambda
Y2 - service time - exponential - \mu
Ym - when the first of an arrival or end of service occurs - Ym = min(Y1, Y2)
P(Ym < y) = P(Y1 < y AND Y2 < y) = F1(y) * F2(y) = exp(-\lambda y) * exp( -\mu y) = exp( -(\lambda + \mu) * y )
Ym is exponentially distributed with mean 1 / (\lambda + \mu)

Birth-death process

A special process where only transitions to neighbouring states are possible That is, if we are in S(j) then the next state can be

S(j-1), a death occurs.
Death rate \mu_j.
S(j+1), a birth occurs
Birth rate \lambda_j.
S(j), neither a birth nor a death occurs.

We now want to examine what happens, in a birth-death process, in the short time between t and t + \Delta t. Use the Poisson distribution

P(exactly one birth ) = (\lambda \Delta t) * exp(-\lambda \Delta t) = \lambda \Delta t + o(\Delta t)
P(exactly one death ) = \mu \Delta t + o(\Delta t)
P(exactly zero births ) = exp(-\lambda \Delta t) = 1 - \lambda \Delta t + o(\Delta t)
P(exactly zero deaths ) = 1 - \mu \Delta t + o(\Delta t)
P(more than one birth and/or death) = o(\Delta t)

o(t) is any function that has the property lim(t->0) o(t)/t = 0.

What happens between t and \Delta t ? Four possibilities

Exactly one birth
- j-1 -> j
- P(n(t+\Delta t) = j AND n(t) = j-1) = \lambda \Delta t + o(\Delta t)
Exactly one death
- j+1 -> j
Neither a birth nor a death
- j -> j
Something else

The book's notation: P(n(t) = j) = pj(t)

Then

p_j(t+dt) = p_j(t) + ( \lambda_j-1 p_j-1(t) + \mu_j+1 p_j+1(t) - (\lambda_j + \mu_j) p_j(t) )dt + o(dt)
(p_j(t+dt) - p_j(t) ) / dt = \lambda_j-1 p_j-1(t) + \mu_j+1 p_j+1(t) - (\lambda_j + \mu_j) p_j(t) + o(dt) / dt
In the limit dt -> 0 dp_j(t) / dt = \lambda_j-1 p_j-1(t) + \mu_j+1 p_j+1(t) - (\lambda_j + \mu_j) p_j(t)

This equation looks as though you could solve it!

Steady State Solutions

Definition of steady state: dp_j(t) / dt = 0.

Then

\lambda_j-1 p_j-1(t) + \mu_j+1 p_j+1(t) - (\lambda_j + \mu_j) p_j(t) = 0

Solve this iteratively

l0 p0 = m1 p1 => p1 = (l0/m1) p0
(l1 + m1) p1 = l0 p0 + m2 p2 => (l1 + m1) (l0/m1) p0 = l0 p0 + m2 p2 => p2 = p0 (l1*l0 / m1*m0)
...
pn = p0 (l_n-1*...*l1*l0 / m_n-1 *...*m1*m0)
Set p0 by the condition sum_n pn = 1.

This looks like you could solve it further, but you can't. How could you possibly solve the differential equation above, in that case?

What do we do? Try simplified examples.

The Simplest Example M/M/1

First M: Markovian (Exponential) birth (interarrival) times

Second M: Markovian (Exponential) life/death (service) times

1: one server

Assumptions:

\lambda_j = \lambda
\mu_j = \mu
Define r = \lambda / \mu

Then

pn = r^n p0
p0 = 1 / (1 + r + r^2 + ...) = 1 - r.
pn = (1 - r) * r^n.

The mean number of jobs in the system is E(r) = sum_n n * (1 - r ) * r^n = r / (1 - r)

Goes to infinity as r -> 1. Why?

Little's law

Mean response time: E(r) = (1/\lambda) * E(n) = 1 / (\mu - \lambda)
Goes to infinity as \lambda -> \mu from below. Why?
What happens when \lambda > \mu?

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