Queueing Theory: Recap

1
Queueing Theory Recap

• Starting point M/M/1
• Poisson arrivals
• Exponential service times
• Markov Chain analysis
• Memoryless property
• Elegant closed-form results
• Key insights into system performance

2
Variations on the M/M/1 Queue

• M/G/1 - generalized service time
• M/D/1 - deterministic service time
• M/M/1/K - finite buffer system
• M/M/c - up to c servers concurrently
• M/M/c/c - Erlang loss model
• M/M/8 - infinite server system
• G/G/1 - generalized arrivals service

3
Even More Variations (1 of 2)

• Balking (discouraged arrivals)
• As the queue becomes longer, new arrivals are
  less likely to join it (e.g., restaurant)
• Aborted jobs (e.g., call center tech support)
• If waiting too long, customers might leave queue
• Variable rate servers
• Service rate changes with time, either randomly,
  or based on load or queue length (e.g., Safeway)
• Vacationing servers
• Server disappears for a while, so that no one
  receives service (e.g., Post Office)

4
Even More Variations (2 of 2)

• Server failures (e.g., power outage)
• Independent failures or catastrophes reduce rate
• Multiple queues vs single shared queue
• Multiple servers, with either separate or shared
  central queue (e.g., bank)
• Jockeying
• Customers can change to a different queue at any
  time (e.g., customs, lane-changing)
• Multi-class priority queues
• Different service classes (e.g., airplane)

5
Queueing Network Models

• So far we have been talking about a queue in
  isolation
• In a queueing network model, there can be
  multiple queues, connected in series or in
  parallel (e.g., CPU, disk, teller)
• Two versions
• Open queueing network models
• Closed queueing network models

6
Open Queueing Network Models

• Assumes that arrivals occur externally from
  outside the system
• Infinite population, with a fixed arrival rate,
  regardless of how many in system
• Unbounded number of customers are permitted
  within the system
• Departures leave the system (forever)

7
Open Queueing Network Example
Jobs In
Jobs Out
8
Closed Queueing Network Models

• Assumes that there is a finite number of
  customers, in a self-contained world
• Finite population arrival rate varies depending
  on how many and where
• Fixed number of customers (N) that recirculate in
  the system (forever)
• Can be analyzed using Mean Value Analysis (MVA)
  and balance equations

9
Closed Queueing Network Example
10
Open Queueing Network Analysis

• Analysis makes use of response time relationship,
  Littles Law, visit ratios, Jacksons Theorem,
  M/M/1 results, etc.
• For a fixed-capacity service center i in an open
  queueing network, the response time Ri is given
  by Ri Si (1 Qi)
• where Ri is the mean response time, Si is
  the mean service time, and Qi is the mean number
  of customers in the queue

11
Closed Queueing Network Analysis

• Self-contained system finite customer
  population, no external inputs/outputs
• Finite population implies that arrival rates at
  different queues depend on the distribution of
  customers in the network
• Analysis makes use of iterative and/or recursive
  solution to compute mean values of performance
  measures (MVA)

12
Mean Value Analysis (MVA)

• A clever analysis technique for closed queueing
  network models (only)
• Provides information about the mean values of
  performance measures (e.g., queue size, response
  time), but not their variance, etc
• The crux of the analysis is given by
• Ri (N) Si (1 Qi (N-1) )
• where Ri is the mean response time for N
  customers, Si is the mean service time, and Qi is
  the mean number of customers in the queue when
  there are N-1 in the system