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Queueing Theory: Recap

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. Variations on the M/M/1 Queue. M/G/1 - generalized service time

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Queueing Theory: Recap

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  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/∞ - 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. Disk A CPU Disk B 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. Disk A CPU Disk B Closed Queueing Network Example

  10. Open Queueing Network Analysis • Analysis makes use of response time relationship, Little’s Law, visit ratios, Jackson’s 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

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