Queueing Theory IE 502: Probabilistic Models Jayendran - - PowerPoint PPT Presentation

Queueing Theory

IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

IE502: Probabilistic Models IEOR @ IITBombay

Typical Configurations

Customer Arrives Customer Departs Server Server Customer Arrives Customer Departs Server Server Customer Departs

Key Attributes Calling population Arrivals pattern Queue capacity; Queue discipline Service times, Capacity

IE502: Probabilistic Models IEOR @ IITBombay

Analytical Models of Queues

• Queueing Theory was developed by

A.K. Erlang in 1909.

– Worked for Copenhagen Telephone Exchange as an engineer, and developed tools to analyze and design telecommunication networks based

• n probability theory.

IE502: Probabilistic Models IEOR @ IITBombay

Standard notation

• Standard notation to identify the main elements that define

the structure of a queueing system is (first suggested by D.G. Kendall, 1953)

M G 1

:

Arrival Process M: Markovian D: Deterministic G: General EK: Erlang Service time distribution M: Markovian D: Deterministic G: General EK: Erlang Number of Servers in Parallel Max capacity (or buffer size)

• f system.

Default : ∞ Size of source Default: ∞ Queue discipline FIFO: First in First Out LIFO: Last in First Out etc Default value: FIFO

IE502: Probabilistic Models IEOR @ IITBombay

M/M/1

• M/M/1 queueing system is a Birth & Death process

characterized by having the arrival rates λ and the service rates µ independent of the state.

• Usual picture of M/M/1 queue
• State space representation of M/M/1 queue
• What is the limiting probabilities Pn for M/M/1?

Pn : long run probability that system contains exactly n customers μ λ

IE502: Probabilistic Models IEOR @ IITBombay

• How do you interpret the figure below?

M/M/1: ρ versus Pi

Pi i ρ=0.2 ρ=0.5 ρ=0.8

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Performance Measures of M/M/1

• We are typically interested in the steady state

performance measures:

– Server Utilization, U – Expected number of customers in the system, L – Expected number of customers in the queue, LQ – Average time a customer spends in the system, W – Average time a customer waits in the queue, WQ

• Before we look at further performance measures
• f M/M/1 queue, let take a look at Little’s Law.

IE502: Probabilistic Models IEOR @ IITBombay

Little’s Law

• First proof published by John D.C. Little (1961)

Little’s Law In steady state, the expected number of entities in the system is equal to the average arrival rate multiplied by the expected time spent in system

– Avg Number in sys = Avg Arrival rate x Avg time in sys – Avg Number in sys = Avg Departure rate x Avg time in sys – Avg Inventory = Avg Throughput rate x Avg Flowtime – Avg WIP = Avg Throughput rate x Avg Flowtime

• Let’s look at an illustrative proof of Little’s Law

IE502: Probabilistic Models IEOR @ IITBombay

Proof of Little’s Law

N(t) t time Arrivals Departures

Number in system

T1 T2 What are the assumptions we have made?

IE502: Probabilistic Models IEOR @ IITBombay

Little’s law in practice

• 1. KanjurMarg ticket counter can serve up to 100

customers per hour. The queue for the counter has 30 people in it. How long will you spend in the queue?

• 2. KReSiT tea shop sells 50 litres of tea per day. The

manager wants the tea to be always fresh, i.e., the tea should not be more than 2 hours old on the average when sold. How much tea should be made & stored? (Assume 10 hr per day)

IE502: Probabilistic Models IEOR @ IITBombay

Coming back to M/M/1 queue

• The steady state performance measures of M/M/1

queue:

– Server Utilization, U – Expected number of customers in the system, L – Expected number of customers in the queue, LQ – Average time a customer spends in the system, W – Average time a customer waits in the queue, WQ