CS457 - System Performance Evaluation - Winter 2008


Questions and Comments

  1. Assignment 4
  2. `Service centre'

Lecture 33

Analytic Queueing Theory - Queueing Networks

Text: Chapters 30--36

Operational Analysis

Closed Networks in Steady State

  1. Can solve only for ratios of the \lambda_i
  2. Cut the system (at an arbitrary location) to define the throughput, X.

B. Interactive System in Steady State

1. Call the set of terminals server 0

2. Throughput

Cut at entrance to terminals: X = \lambda_0

Define visit ratio: number of visits to service i per job

Why did we define the visit ratio?

3. Utilization

U_i = \lambda_i s_i = X V_i s_i = X D_i

4. Upper bound on throughput

D = \sum D_i

X(n): throughput with N users

5. Bottleneck analysis

  1. b - server with highest demand
  2. D_b = max{D_i} = D_max
  3. U_b = X D_b <= 1
  4. X <= 1/D_max

Therefore, X(N) <= min( 1/D_max, N / (D+Z) )

6. Lower bound on mean response time

  1. Treat entire system as one server. Then R(N) = N/X - Z
  2. R(N) >= D = sum_i D_i because R(N) <= N / (D+Z)
  3. X(N) <= 1 / D_max, so that R(N) <= ND_max - Z


Markovian Analysis

Assumptions

  1. All random variables Markovian (exponentially distributed)
  2. Solutions are product form

Solution Method (with M/M/1 example)

  1. Draw state transition diagram
  2. Develop balance equations
  3. Assume product form solution
  4. Verify that product form solution satisfies balance equations
  5. Evaluate any constants
  6. Determine performance metrics

Challenging Example: General Open Network

State is (n_1, n_2, ... , n_M).

Connectivity (conservation) equations are

1. State transition diagram

  1. (n_1, ..., n_i - 1, ..., n_M) -> (n_1, ..., n_i, ..., n_M): \gamma_i
  2. (n_1, ..., n_i + 1, ..., n_M) -> (n_1, ..., n_i, ..., n_M): \mu_(i+1) p_(i+1)(M+1)
  3. (n_1, ..., n_i - 1, ..., n_j + 1, ..., n_M) -> (n_1, ..., n_i, ..., n_M): \mu_j p_ji
  4. (n_1, ..., n_i, ..., n_M) -> (n_1, ..., n_i, ..., n_M): \mu_i p_ii
  5. (n_1, ..., n_i, ..., n_M) -> (n_1, ..., n_i + 1, ..., n_M): \gamma_i
  6. (n_1, ..., n_i, ..., n_M) -> (n_1, ..., n_i - 1, ..., n_M): \mu_i p_i(M+1)
  7. (n_1, ..., n_i, ..., n_M) -> (n_1, ..., n_i - 1, ..., n_j + 1, ...,, n_M): \mu_i p_ij

2. Balance equations


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