1. Introduction to Queuing Theory

2. Queueing theory definitions
(Bose) “the basic phenomenon of queueing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers.”
(Wolff) “The primary tool for studying these problems [of congestions] is known as queuing theory.”
(Kleinrock) “We study the phenomena of standing, waiting, and serving, and we call this study Queuing Theory." "Any system in which arrivals place demands upon a finite capacity resource may be termed a queueing system.”
(Mathworld) “The study of the waiting times, lengths, and other properties of queues.”

3. Applications of Queuing Theory
Telecommunications
Traffic control
Determining sequence of computer operations
Predicting computer performance
Health services (e.g., control of hospital bed assignments)
Airport traffic, airline ticket sales
Layout of manufacturing systems.

4. Example application of queuing theory
In many retail stores and banks
multiple line/multiple checkout system  a queuing system where customers wait for the next available cashier
We can prove using queuing theory that : throughput improves increases when queues are used instead of separate lines

5. Example application of queuing theory

6. Queuing theory for studying networks
View network as collections of queues
FIFO data-structures
Queuing theory provides probabilistic analysis of these queues
Examples:
Average length
Average waiting time
Probability queue is at a certain length
Probability a packet will be lost

7. Model Queuing System Use Queuing models to
Server System
Queuing System
Queue
Server
Use Queuing models to
Describe the behavior of queuing systems
Evaluate system performance

8. Characteristics of queuing systems
Arrival Process
The distribution that determines how the tasks arrives in the system.
Service Process
The distribution that determines the task processing time
Number of Servers
Total number of servers available to process the tasks

9. Kendall Notation 1/2/3(/4/5/6)
Six parameters in shorthand
First three typically used, unless specified
Arrival Distribution
Service Distribution
Number of servers
Total Capacity (infinite if not specified)
Population Size (infinite)
Service Discipline (FCFS/FIFO)

10. Distributions
M: stands for "Markovian", 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)

11. Kendall Notation Examples
M/M/1:
Poisson arrivals and exponential service, 1 server, infinite capacity and population, FCFS (FIFO)
the simplest ‘realistic’ queue
M/M/m
Same, but M servers
G/G/3/20/1500/SPF
General arrival and service distributions, 3 servers, 17 queue slots (=20-3), 1500 total jobs, Shortest Packet First

12. Analysis of M/M/1 queue Given: Solve:
l: Arrival rate of jobs (packets on input link)
m: Service rate of the server (output link)
Solve:
L: average number in queuing system
Lq average number in the queue
W: average waiting time in whole system
Wq average waiting time in the queue

13. M/M/1 queue model L Lq l m Wq W

14. Little’s Law
System
Arrivals
Departures
Little’s Law: Mean number tasks in system = mean arrival rate x mean response time
Observed before, Little was first to prove
Applies to any system in equilibrium, as long as nothing in black box is creating or destroying tasks

15. Proving Little’s Law J = Shaded area = 9 Same in all cases!
# in
System 1 2 3
Arrivals 1 2 3
Packet #
Departures
Time
1 2 3
Time in
System
Packet # 1 2 3
Time
J = Shaded area = 9
Same in all cases!

16. Definitions J: “Area” from previous slide N: Number of jobs (packets)
T: Total time
l: Average arrival rate
N/T
W: Average time job is in the system
= J/N
L: Average number of jobs in the system
= J/T

17. Proof: Method 1: Definition
1 2 3
Time in
System
(W)
Packet # (N) 1 2 3 1 2 3
# in
System
(L) =
Time (T)

18. Proof: Method 2: Substitution
Tautology

19. M/M/1 queue model L Lq l m Wq W
L=λW
Lq=λWq

20. Poisson Process
For a poisson process with average arrival rate , the probability of seeing n arrivals in time interval delta t

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

22. M/M/1 queue model l m Wq W L Lq
L=λW
Lq=λWq
W = Wq + (1/μ)

23. Solving queuing systems
4 unknowns: L, Lq W, Wq
Relationships:
L=lW
Lq=lWq
W = Wq + (1/m)
If we know any 1, can find the others

24. Analysis of M/M/1 queue
Goal: A closed form expression of the probability of the number of jobs in the queue (Pi) given only l and m

25. Equilibrium conditions
Define to be the probability of having n tasks in the system at time t

26. Equilibrium conditions

27. Solving for P0 and Pn
Step 1
Step 2

28. Solving for P0 and Pn
Step 3
Step 4

29. Solving for L

30. Solving W, Wq and Lq

31. Online M/M/1 animation

32. Response Time vs. Arrivals

33. Stable Region
linear region

34. Example
On a network gateway, measurements show that the packets arrive at a mean rate of 125 packets per second (pps) and the gateway takes about 2 millisecs to forward them. Assuming an M/M/1 model, what is the probability of buffer overflow if the gateway had only 13 buffers. How many buffers are needed to keep packet loss below one packet per million?

35. Example Measurement of a network gateway:
mean arrival rate (l): 125 Packets/s
mean response time (m): 2 ms
Assuming exponential arrivals:
What is the gateway’s utilization?
What is the probability of n packets in the gateway?
mean number of packets in the gateway?
The number of buffers so P(overflow) is <10-6?

36. Example Arrival rate λ = Service rate μ =
Gateway utilization ρ = λ/μ =
Prob. of n packets in gateway =
Mean number of packets in gateway =

37. Example Arrival rate λ = 125 pps Service rate μ = 1/0.002 = 500 pps
Gateway utilization ρ = λ/μ = 0.25
Prob. of n packets in gateway =
Mean number of packets in gateway =

38. Example Probability of buffer overflow:
To limit the probability of loss to less than 10-6:

39. Example
Probability of buffer overflow: = P(more than 13 packets in gateway)
To limit the probability of loss to less than 10-6:

40. Example
Probability of buffer overflow: = P(more than 13 packets in gateway) = ρ13 = = 1.49x = 15 packets per billion packets
To limit the probability of loss to less than 10-6:

41. Example
Probability of buffer overflow: = P(more than 13 packets in gateway) = ρ13 = = 1.49x = 15 packets per billion packets
To limit the probability of loss to less than 10-6:

42. Example
To limit the probability of loss to less than 10-6: or

43. Example To limit the probability of loss to less than 10-6:
or = ≈10 buffers

44. Example One fast server vs. m slow servers?
In terms of delay, what will happen?

45. Example
Customers arrive at a fast-food restaurant at a rate of 5/minute and wait to receive their order for an average of 5 minutes. Customers eat in the restaurant with probability 0.5 and carry out their order without eating with probability 0.5. A meal requires an average of 20 minutes. What is the average number of customers in the restaurant?

46. Example
Empty taxis pass by a street comer at a Poisson rate of 2 per minute and pick up a passenger if one is waiting there. Passengers arrive at the street corner at a Poisson rate of 1 per minute and wait for a taxi only if there are fewer than 4 persons waiting; otherwise, they leave and never return. Find the average waiting time of a passenger who joins the queue.

47. Example
The average time,T, a car spends in a certain traffic system is related to the average number of cars N in the system by a relation of the form T = a + bN2, where a > 0, b > 0 are give in scalars.
What is the maximal car arrival rate that the system can sustain?

48. Example
A person enters a bank and finds all of the four clerks busy serving customers. There are no other customers in the bank, so the person will start service as soon as one of the customers in service leaves. Customers have independent, identical, exponential distribution of service time.
What is the probability that the person will be the last to leave the bank assuming that no other customers arrive?
If the average service time is 1 minute, what is the average time the person will spend?