### Queuing Theory 1 Notes

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Lecture 11: Queuing Theory 1

• Introduction • Taha (Chapter 15) • Life cycles of queues
• Key Factors Influencing the Life Cycles of Queues • What is the system?
• What is "the System"? • Arrival patterns
• Arrival Patterns • Queue discipline
• Queue Discipline • Service characteristics
• Service Characteristics • Queue development
• Queue Development • Markov chains
• Queues and Markov Chains • Steady states, balanced equations and recurrence relations
• General Equations of the Steady State • Case 1: Rates are dependent on k (the number in the system)
• Single-Server Rates-Independent Model: Major Statistics • Case 2: Rates independent of k (only one server)
• Example • Major statistics of case 2

Definitions
• Balance Equation for state 0 = In state 0 in the steady state, the mean
Introduction entering rate balances the mean leaving rate
• Any sequence of arrivals and departures is a realisation of a stochastic process • Initial Phase = When the shop opens and for a short period after that
• The queuing system may exhibit steady state or equilibrium behaviour • Steady State = When the various probabilities have become constant
• Transient Phase = When the probability that there are j customers in
Key Factors Influencing the Life Cycles of Queues the system varies with time t
• Three key factors influence the life cycles of individual queues:
a. The arrival patterns of the 'items' Formulae
b. The logic of the queue behaviour •
c. The characteristics of the service facility •

• Almost all queuing systems are stochastic processes •
• We look to identify the expected values of the system characteristics • E[inter-arrive time]=1/
○ Not to predict the future • General Balance Equation for state j =

What is "the System"? • Number in the system = Number being served + number queuing
• The system consists only of the customers who are being served or who are queuing • P(j in the system) is:
• The number in the system does not include: • P(system is in state 0 at time t and an arrival occurs in (t,t+dt)) =
○ The servers
○ The customers who are still shopping • P(system switches from state 1 to state 2 in (t,t+dt)) =
• Number in the system = Number being served + number queuing
• Probability of x arrivals per unit time:
Arrival Patterns • The cumulative density function of arrival pattern:
• 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 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
 We then write simply as
• 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

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