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Queuing Theory 1 Notes

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Lecture 11: Queuing Theory 1 Topics

• Introduction

• Key Factors Influencing the Life Cycles of Queues

• What is "the System"?

• Arrival Patterns

• Queue Discipline

• Service Characteristics

• Queue Development

• Queues and Markov Chains

• General Equations of the Steady State

• Single-Server Rates-Independent Model: Major Statistics

• Example

Reading

• Taha (Chapter 15)

Definitions

• Balance Equation for state 0 = In state 0 in the steady state, the mean entering rate balances the mean leaving rate

• Initial Phase = When the shop opens and for a short period after that

• Steady State = When the various probabilities have become constant

• Transient Phase = When the probability that there are j customers in the system varies with time t

Introduction

• Any sequence of arrivals and departures is a realisation of a stochastic process

• The queuing system may exhibit steady state or equilibrium behaviour Key Factors Influencing the Life Cycles of Queues

• Three key factors influence the life cycles of individual queues: a. The arrival patterns of the 'items' b. The logic of the queue behaviour c. The characteristics of the service facility

Formulae

• E[inter-arrive time]=1/

• General Balance Equation for state j =

• Almost all queuing systems are stochastic processes

• We look to identify the expected values of the system characteristics
○ Not to predict the future

• Number in the system = Number being served + number queuing

What is "the System"?

• The system consists only of the customers who are being served or who are queuing

• The number in the system does not include:
○ The servers
○ The customers who are still shopping

• Number in the system = Number being served + number queuing Arrival Patterns


is the average number of arrivals per unit of time


is the probability of an arrival in the extremely small time interval (t,t+dt)
○ This is independent of what happened earlier

• These arrivals follow a Poisson distribution with an expected rate of per unit time
○ The time between successive arrivals has an exponential distribution with mean value

• P(j in the system) is:

• P(system is in state 0 at time t and an arrival occurs in (t,t+dt)) =

• P(system switches from state 1 to state 2 in (t,t+dt)) =

• Probability of x arrivals per unit time:

• The cumulative density function of arrival pattern:

units of time

• Probability of x arrivals per unit time:

The cumulative density function of arrival pattern: The inter-arrival times have an exponential distribution E[inter-arrive time]=1/
We assume:
○ that the arrivals are independent (∴ individual)
○ The arrivals are random in time
○ The arrival rate does not vary with time

Queue Discipline

• i.e. FIFO, LIFO, random, balking, jockeying, reserving, swapping, priorities

• There may be limits on queue size

• Customers may leave the queue after a certain time if they haven't been served Service Characteristics

• There may be more than one server

• There may be specialist servers i.e. less than 10 items or cash only

• We assume service times have an exponential distribution
○ Expected number of services per unit time being
○ There is a constant probability that a service will end during the time period
○ A truncated Normal or preferably a Beta distribution is more likely Queue Development

• There are usually 3 development phases
○ Initial Phase = When the shop opens and for a short period after that
○ Transient Phase = When the probability that there are j customers in the system varies with time t

○ Steady State = When the various probabilities have become constant simply as
 We then write

• The first two phases are often bundled together for convenience

• We are most interested in the steady state Queues and Markov Chains

• Some queues can be represented as Markov Chains

• We suppose arrivals and departures occur singly at discrete times

• We also assume that the probability of an individual arrival or departure is independent of what has happened previously

• We model this as a Markov Process which can be represented by a Markov Chain

• We can also consider queues in continuous time General Equations of the Steady State Case 1: Rates are Dependent on k, the No. of Customers in the System
○ Let arrival rate be
 This is either an increasing function of j (e.g. customers being attracted to a successful restaurant)
 Or a decreasing function of j (e.g. customers entering a supermarket may balk (leave) if they observe that long queues are building up)
○ The values of (from ) are all different We also assume that the potential service rates also vary with j

Course Notes Page 22

Key Points

• Life cycles of queues

• What is the system?

• Arrival patterns

• Queue discipline

• Service characteristics

• Queue development

• Markov chains

• Steady states, balanced equations and recurrence relations

• Case 1: Rates are dependent on k (the number in the system)

• Case 2: Rates independent of k (only one server)

• Major statistics of case 2

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