Queueing Theory

Airline Industry (routing and flight plans, crew scheduling, revenue management)

Telecommunications (network routing, queue control)

Manufacturing Industry (system throughput and bottleneck analysis, inventory control, production scheduling, capacity planning)

Healthcare (hospital management, facility design) Transportation (traffic control, logistics, network

flow, airport terminal layout, location planning)

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An area of interest in Operations Research is the analysis of stochastic processes (i.e., processes with random variability), relies on results from applied probability and statistical

modeling. Many real-world problems involve uncertainty, and

mathematics has been extremely useful in identifying ways to manage it.

Modeling uncertainty is important in risk analysis for complex systems, such as space shuttle flights large dam operations nuclear power generation

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Related to the topic of stochastic processes is queueing theory (i.e., the analysis of waiting lines).

A common example is the single-server queue in which customer arrivals and service times are random..

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Mathematical analysis has been essential in understanding queue behavior and quantifying impacts of decisions.

Equations have been derived for the queue length, waiting times, and probability of no delay, and other measures.

The results have applications in many types of queues, such as customers at a bank or supermarket checkout orders waiting for production ships docking at a harbor users of the internet customers served at a restaurant.

Every day life: waiting in queues to make a bank deposit, pay toll, pay for groceries, mail a package, obtain food in cafeteria, meet physician

We spend 37 billion hours in waiting each year. Making machines wait to be repaired results in

lost production. Waiting for take-off or landing can disrupt schedules. Delaying service jobs beyond due dates may result in lost future business.

How to operate queueing system in most effective way; balance between cost of service and the amount of waiting

Queuing System

Queue Server

Customers

For a single-server queue in which customer arrivals and service times are random, the figure illustrates the queue, and the curve shows how sensitive the average queue length becomes under high traffic intensity conditions.

Time

Time

Arrival event

Delay

Begin service

Begin service

Arrival event

Delay

Activity

Activity

End service

End service

Customer n+1

Customer n

Interarrival

Kendall Notation 1/2/3(/4/5/6)1. Arrival Distribution2. Service Distribution3. Number of servers 4. Total storage (including servers)5. Population Size6. Service Discipline (FIFO)

M: "Markovian / Poisson" , implying exponential distribution

for service times or inter-arrival times.

D: Deterministic (e.g. fixed

constant)

Ek: Erlang with parameter k

Hk: Hyperexponential with param. k

G: General (anything)

Inter-arrival time t (time between arrivals) in a Poisson process follows exponential distribution with parameter (mean)

1)(

)(

tE

etP t

fT(t)

t1

)( TE

M/M/1: Poisson arrivals and exponential

service, 1 server, infinite capacity and population, FCFS (FIFO)

the simplest ‘realistic’ queue M/M/m/m

Same, but m servers, m storage (including servers) Ex: telephone

Given:

: Arrival rate (mean) of customers (jobs)

: Service rate (mean) of the server

: Traffic intensity factor

Solve: L: average number in queuing system Lq average number in the queue ~ “1” W: average waiting time in whole system Wq average waiting time in the queue ~

“1/”

1

Wq

W

L

Lq

M/M/1 exponentially distributed inter-arrival times and service times and single server (FIFO queue discipline)

L = Lq = L-2 W = Wq = Pn = n

L = W Lq = Wq (Little’s Law) The average number of customers in a stable

system (over some time interval) is equal to their average arrival rate, multiplied by their average time in the system.

Imagine a small shop with a single counter and an area for browsing, where only one person can be at the counter at a time, and no one leaves without buying something. So the system is roughly:

Entrance → Browsing → Counter → Exit This is a stable system, so the rate at which

people enter the store is the rate at which they arrive at the counter and the rate at which they exit as well. We call this the arrival rate.

Little's Law tells us that the average number of customers in the store is the arrival rate times the average time that a customer spends in the store.

Assume customers arrive at the rate of 10 per hour and stay an average of 0.5 hour. This means we should find the average number of customers in the store at any time to be 5.

Now suppose the store is considering doing more advertising to raise the arrival rate to 20 per hour. The store must either be prepared to host an average of 10 occupants or must reduce the time each customer spends in the store to 0.25 hour.

The store might achieve the latter by ringing up the bill faster or by walking up to customers who seem to be taking their time browsing and saying, "Can I help you?".

We can apply Little's Law to systems within the shop. For example, the counter and its queue. Assume we notice that there are on average 2 customers in the queue and at the counter. We know the arrival rate is 10 per hour, so customers must be spending 0.2 hour on average checking out.

We can even apply Little's Law to the counter itself. The average number of people at the counter would be in the range (0,1) since no more than one person can be at the counter at a time. In that case, the average number of people at the counter is also known as the counter's utilization.

1W

Waiting vs. Utilization

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1 1.2

W(s

ec)

On a network router, measurements show the packets arrive at a mean rate of

125 packets per second (pps) the router takes about 2 millisecs to

forward a packet Assuming an M/M/1 model What is the probability of buffer

overflow if the router had only 13 buffers

How many buffers are needed to keep packet loss below one packet per million?

Arrival rate λ = 125 pps Service rate μ = 1/0.002 = 500 pps Router utilization ρ = λ/μ = 0.25 Prob. of n packets in router =

Mean number of packets in router =

nn )25.0(75.0ρ)ρ1(

33.057.0

25.0

ρ1

ρ

Example

Example

Probability of buffer overflow: = P(more than 13 packets in router) = ρ13 = 0.2513 = 1.49x10-8

= 15 packets per billion packets To limit the probability of loss to less

than 10-6:

= 9.96

610ρ n

25.0log/10log 6n