1 / 12

Understanding Queueing Models: Performance Metrics and Markov Decision Processes

This document explores essential topics in queueing theory, focusing on performance measures such as waiting time, number of customers, and system utilization. It covers various queue models including M/M/1 and incorporates concepts like arrival rates and service rates. Key performance metrics like Lq (average number in queue) and Wq (average waiting time) are derived from Little’s Law, highlighting the effects of server utilization. Additionally, it discusses factors often ignored in basic models, such as rush-hour effects, batching, and queue pooling.

shannon-christian
Télécharger la présentation

Understanding Queueing Models: Performance Metrics and Markov Decision Processes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Previously • Optimization • Probability Review • Inventory Models • Markov Decision Processes

  2. Agenda • Queues

  3. T time in system Tq waiting time (time in queue) N #customers in system Nq #customers in queue W = E[T] Wq= E[Tq] L = E[N] Lq= E[Nq]  fraction of time servers are busy (utilization) departures arrivals queue servers system Performance Measures

  4. Plain-Vanilla Queue • 1 queue, • 1 class of customers • identical servers • time between customer arrivals is independent • mean rate of arrivals, rate of serviceconstant

  5. What Are We Ignoring? • Rush-hour effects • Priority classes • Balking, Reneging, Jockeying • Batching • Multi-step processes • Queue capacity

  6. Parameters •  mean arrival rate of customers (per unit time) • c servers • µ mean service rate (per unit time)

  7. Some Relations •  = /(cµ) utilization • c = /µ average # of busy servers • W = Wq + 1/µ • L = Lq + c • Little’s Law:Lq =  Wq and L =  W

  8. Lq  So What is Lq? • Depends on details. M/M/1 queue (exponential arrival times, exponential processing times, 1 server)

  9. Qualitatively • Dependence on  1 means Lq • Increased variability (arrival / service times) • Lq increases • Pooling queues • Lq decreases

  10. number of servers: 1, 2, … distribution of the time between arrivals distribution of the processing time Queue Notation M/M/1 D/M/1 M/G/3 … M / M / 1 M = ‘Markov’ exponential distribution D = ‘Deterministic’ constant G = ‘General’ other

  11. Back to Lq… • Lq=E[Nq] • M/G/1 • 2 = variance of the service time • M/D/1 • M/M/1

  12. M/M/1 • N+1 has distribution Geometric(1-) • Var[Nq] = (1+-2) 2/(1-)2 • StDev[Nq] > E[Nq] /  • Pooling queues decreases Lq Ex. (p501) 3 clinics with 1 nurse each (M/M/1) =4/hr, µ=1/13min =87%, Lq=5.6, Wq=1.4hr Consolidated (M/M/3) =12/hr, µ=60/13hr, =87% Lq=4.9, Wq=0.4

More Related