����Dr. K. KRISHNAN Ph.D.�ASSISTANT PROFESSOR & RESEARCH SUPERVISOR�C. P. A. COLLEGE – BODINAYAKANUR�����INTRODUCTION TO �QUEUING SYSTEMS�
17. 12. 2021
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Overview
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What is Queuing Theory?
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Prototype Example – ER at County Hospital
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Why is Queuing Analysis Important?
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Process capacity
Cost
Cost of waiting
Cost of
service
Total
cost
A Cost/Capacity Tradeoff Model
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Examples of Real World Queuing Systems?
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Components of a Basic Queuing Process
Calling Population
Queue
Service Mechanism
Input Source
The Queuing System
Jobs
Arrival Process
Queue Configuration
Queue Discipline
Served Jobs
Service Process
leave the system
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Components of a Basic Queuing Process
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Components of a Basic Queuing Process
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Mitigating Effects of Long Queues
Customer Behavior
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server
A Commonly Seen Queuing Model
C C C … C
Customers (C)
Customer =C
The Queuing System
The Queue
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A Commonly Seen Queuing Model
Definition: A stochastic (or random) variable T∈exp(α ), i.e., is exponentially distributed with parameter α, if its frequency function is:
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The Exponential Distribution and Queuing
⇒ The Cumulative Distribution Function is:
⇒ The mean = E[T] = 1/α
⇒ The Variance = Var[T] = 1/ α2
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The Exponential Distribution
Time between arrivals
Mean=
E[T]=1/α
Probability density
t
fT(t)
α
Two equivalent definitions of the Poisson Process
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The Poisson Process
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a) Aggregation of N Poisson processes with intensities
{λ1, λ2, …, λn} renders a new Poisson process with intensity λ= λ1+ λ2+…+ λn.
b) Disaggregating a Poisson process X(t)∈Po(λt) into N sub-processes {X1(t), X2(t), , …, X3(t)} (for example N customer types) where Xi(t) ∈Po(λit) can be done if
– For every arrival the probability of belonging to sub-process i = pi
– p1+ p2+…+ pN = 1, and λi = pi λ
Properties of the Poisson Process
N(t) = Number of customers/jobs in the system at time t
Pn(t) = The probability that at time t, there are n customers/jobs in the system.
λn = Average arrival intensity (= # arrivals per time unit) at n customers/jobs in the system
μn = Average service intensity for the system when there are n customers/jobs in it. (Note, the total service intensity for all occupied servers)
ρ = The utilization factor for the service facility. (= The expected fraction of the time that the service facility is being used)
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Terminology and Notation
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Example – Service Utilization Factor
⇒
⇒
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Queuing Theory Focus on Steady State
With few exceptions Queuing Theory has focused on analyzing steady state behavior
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Transient and Steady State Conditions
Pn = The probability that there are exactly n customers/jobs in the system (in steady state, i.e., when t→∞)
Ls = Expected number of customers in the system (in steady state)
Lq = Expected number of customers in the queue (in steady state)
Ws = Expected waiting time of customer in the system
Wq= Expected waiting time of customer in the queue
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Notation For Steady State Analysis
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Little’s Formula
Assumptions
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Birth-and-Death Processes
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A Birth-and-Death Process Rate Diagram
λ0
λ1
λn-1
λn
0
1
2
n-1
n
n+1
μ1
μ2
μn
μn+1
n
= State n, i.e., the case of n customers/jobs in the system
Assumptions - the Basic Queuing Process
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The M/M/1 - model
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Example – ER at Hospital
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Summary of Results – County Hospital
Characteristic | One doctor (c=1) | Two Doctors (c=2) |
ρ | 2/3 | 1/3 |
P0 | 1/3 | 1/2 |
(1-P0) | 2/3 | 1/2 |
P1 | 2/9 | 1/3 |
Lq | 4/3 patients | 1/12 patients |
Ls | 2 patients | 3/4 patients |
Wq | 2/3 h = 40 minutes | 1/24 h = 2.5 minutes |
Ws | 1 h | 3/8 h = 22.5 minutes |
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Analyzing Linear Waiting Costs
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SC = c*CS(μ)
Analyzing Service Costs
Determining μ and c
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A Decision Model for System Design
From a structural point of view, a few fast servers are usually better than several slow ones with the same maximum capacity
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Analyzing Design-Cost Tradeoffs
Min TC = WC + SC
Process capacity
Cost
WC
SC
TC
����Thank You�
Questions Please ?
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