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Introduction to Operations Research

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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)

- 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

- 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..

- 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

Customers

Server

Queue

Queuing System

- 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.

Customer n

Arrival event

Begin service

End service

Delay

Activity

Time

Interarrival

Arrival event

Begin service

End service

Delay

Activity

Time

Customer n+1

Kendall Notation 1/2/3(/4/5/6)

- Arrival Distribution
- Service Distribution
- Number of servers
- Total storage (including servers)
- Population Size
- 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)

fT(t)

t

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

- 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:
l: Arrival rate (mean) of customers (jobs)

m: Service rate (mean) of the server

r = l/m: 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/m”

L

Lq

l

m

Wq

W

- M/M/1 exponentially distributed inter-arrival times and service times and single server (FIFO queue discipline)
- L = l/(m-l)Lq = L-r = r2/(1-r) = l2/m(m-l)
- W = 1/(m-l)Wq = l/m(m-l)Pn = (1-r)rn
- L = lWLq = lWq(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.

- 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?

Example

- 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 =

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