1. CPE 619 Introduction to Queuing Theory
Aleksandar Milenković
The LaCASA Laboratory
Electrical and Computer Engineering Department
The University of Alabama in Huntsville

2. Overview Queueing Notation Rules for All Queues Little's Law
Types of Stochastic Processes

3. Queueing Models: What Will You Learn?
What are various types of queues?
What is meant by an M/M/m/B/K queue?
How to obtain response time, queue lengths, and server utilizations?
How to represent a system using a network of several queues?
How to analyze simple queueing networks?
How to obtain bounds on the system performance using queueing models?
How to obtain variance and other statistics on system performance?
How to subdivide a large queueing network model and solve it?

4. Basic Components of a Queue
2. Service time
distribution
1. Arrival
process
6. Service
discipline
5. Customer Population
4. Waiting
positions
3. Number of
servers
Example: students at a typical computer terminal room with a number of terminals. If all terminals are busy, the arriving students wait in a queue.

5. Kendall Notation A/S/m/B/K/SD
A: Arrival process
S: Service time distribution
m: Number of servers
B: Number of buffers (system capacity)
K: Population size, and
SD: Service discipline

6. Arrival Process Arrival times: Interarrival times:
tj form a sequence of Independent and Identically Distributed (IID) random variables
The most common arrival process: Poisson arrivals
Inter-arrival times are exponential + IID  Poisson arrivals
Notation:
M = Memoryless = Poisson
E = Erlang
H = Hyper-exponential
G = General  Results valid for all distributions

7. Service Time Distribution
Time each student spends at the terminal
Service times are IID
Distribution: M, E, H, or G
Device = Service center = Queue
Buffer = Waiting positions

8. Service Disciplines First-Come-First-Served (FCFS)
Last-Come-First-Served (LCFS)
Last-Come-First-Served with Preempt and Resume (LCFS-PR)
Round-Robin (RR) with a fixed quantum.
Small Quantum  Processor Sharing (PS)
Infinite Server: (IS) = fixed delay
Shortest Processing Time first (SPT)
Shortest Remaining Processing Time first (SRPT)
Shortest Expected Processing Time first (SEPT)
Shortest Expected Remaining Processing Time first (SERPT).
Biggest-In-First-Served (BIFS)
Loudest-Voice-First-Served (LVFS)

9. Common Distributions M: Exponential Ek: Erlang with parameter k
Hk: Hyper-exponential with parameter k
D: Deterministic  constant
G: General  All
Memoryless:
Expected time to the next arrival is always 1/l regardless of the time since the last arrival
Remembering the past history does not help

10. Example: M/M/3/20/1500/FCFS
Time between successive arrivals is exponentially distributed
Service times are exponentially distributed
Three servers
20 Buffers = 3 service + 17 waiting
After 20, all arriving jobs are lost
Total of 1500 jobs that can be serviced
Service discipline is first-come-first-served
Defaults:
(Only the first 3 parameters are sufficient to indicate the type)
Infinite buffer capacity
Infinite population size
FCFS service discipline
G/G/1 = G/G/1/1/1/FCFS

11. Group Arrivals/Service
Bulk arrivals/service
M[x]: x represents the group size
G[x]: a bulk arrival or service process with general inter-group times
Examples:
M[x]/M/1 : Single server queue with bulk Poisson arrivals and exponential service times
M/G[x]/m: Poisson arrival process, bulk service with general service time distribution, and m servers

12. Key Variables 1 2 m Previous Arrival Arrival Begin Service End Servicet w s r n nq ns l
Time

13. Key Variables (cont’d)
t = Inter-arrival time = time between two successive arrivals
l = Mean arrival rate = 1/E[t] May be a function of the state of the system, e.g., number of jobs already in the system
s = Service time per job
m = Mean service rate per server = 1/E[s]
Total service rate for m servers is mm
n = Number of jobs in the system. This is also called queue length
Note: Queue length includes jobs currently receiving service as well as those waiting in the queue

14. Key Variables (cont’d)
nq = Number of jobs waiting
ns = Number of jobs receiving service
r = Response time or the time in the system = time waiting + time receiving service
w = Waiting time = Time between arrival and beginning of service

15. Rules for All Queues Rules: The following apply to G/G/m queues
1. Stability Condition: l < mm Finite-population and the finite-buffer systems are always stable
2. Number in System versus Number in Queue: n = nq+ ns Notice that n, nq, and ns are random variables. E[n]=E[nq]+E[ns] If the service rate is independent of the number in the queue,
Cov(nq,ns) = 0

16. Rules for All Queues (cont’d)
3. Number versus Time: If jobs are not lost due to insufficient buffers, Mean number of jobs in the system = Arrival rate  Mean response time
4. Similarly, Mean number of jobs in the queue = Arrival rate  Mean waiting time
This is known as Little's law.
5. Time in System versus Time in Queue r = w + s r, w, and s are random variables
E[r] = E[w] + E[s]

17. Rules for All Queues (cont’d)
6. If the service rate is independent
of the number of jobs in the queue, Cov(w,s)=0

18. Little's Law Black Box Departures Arrivals
Mean number in the system = Arrival rate  Mean response time
This relationship applies to all systems or parts of systems in which the number of jobs entering the system is equal to those completing service
Named after Little (1961)
Based on a black-box view of the system:
In systems in which some jobs are lost due to finite buffers, the law can be applied to the part of the system consisting of the waiting and serving positions
Black Box
Arrivals
Departures

19. Proof of Little's Law 1 2 3 4 5 6 7 8 Job number Arrival Departure 4
Number in System 3 2 1 1 2 3 4 5 6 7 8
Time
Time
If T is large, arrivals = departures = N
Arrival rate = Total arrivals/Total time= N/T
Hatched areas = total time spent inside the system by all jobs = J
Mean time in the system= J/N
Mean Number in the system = J/T = (N/T) (J/N) = Arrival rate  Mean time in the system 1 2 3 4
Time in System
Job number

20. Application of Little's Law
Arrivals
Departures
Applying to just the waiting facility of a service center
Mean number in the queue = Arrival rate  Mean waiting time
Similarly, for those currently receiving the service, we have:
Mean number in service = Arrival rate  Mean service time

21. Example 30.3
A monitor on a disk server showed that the average time to satisfy an I/O request was 100 milliseconds. The I/O rate was about 100 requests per second. What was the mean number of requests at the disk server?
Using Little's law:
Mean number in the disk server
= Arrival rate  Response time
= 100 (requests/second) (0.1 seconds)
= 10 requests

22. Stochastic Processes Process: Function of time
Stochastic Process: Random variables, which are functions of time
Example 1:
n(t) = number of jobs at the CPU of a computer system
Take several identical systems and observe n(t)
The number n(t) is a random variable
Can find the probability distribution functions for n(t) at each possible value of t
Example 2:
w(t) = waiting time in a queue

23. Types of Stochastic Processes
Discrete or Continuous State Processes
Markov Processes
Birth-death Processes
Poisson Processes

24. Discrete/Continuous State Processes
Discrete = Finite or Countable
Number of jobs in a system n(t) = 0, 1, 2, ....
n(t) is a discrete state process
The waiting time w(t) is a continuous state process
Stochastic Chain: discrete state stochastic process

25. Markov Processes
Future states are independent of the past and depend only on the present
Named after A. A. Markov who defined and analyzed them in 1907
Markov Chain: discrete state Markov process
Markov  It is not necessary to know how long the process has been in the current state  State time has a memoryless (exponential) distribution
M/M/m queues can be modeled using Markov processes
The time spent by a job in such a queue is a Markov process and the number of jobs in the queue is a Markov chain

26. Birth-Death Processes1 2
j-1 j
j+1 … l0 l1 l2
lj-1 lj
lj+1 m1 m2 m3 mj
j+1
 j+2
The discrete space Markov processes in which the transitions are restricted to neighboring states
Process in state n can change only to state n+1 or n-1.
Example: the number of jobs in a queue with a single server and individual arrivals (not bulk arrivals)

27. Poisson Processes
Interarrival time s = IID and exponential  number of arrivals n over a given interval (t, t+x) has a Poisson distribution  arrival = Poisson process or Poisson stream
Properties:
1.Merging:
2.Splitting: If the probability of a job going to ith substream is pi, each substream is also Poisson with a mean rate of pi l

28. Poisson Processes (cont’d)
3.If the arrivals to a single server with exponential service time are Poisson with mean rate l, the departures are also Poisson with the same rate l provided l < m

29. Poisson Process (cont’d)
4. If the arrivals to a service facility with m service centers are Poisson with a mean rate l, the departures also constitute a Poisson stream with the same rate l, provided l < åi mi. Here, the servers are assumed to have exponentially distributed service times.

30. Relationship Among Stochastic Processes

31. Summary Kendall Notation: A/S/m/B/k/SD, M/M/1
Number in system, queue, waiting, service Service rate, arrival rate, response time, waiting time, service time
Little’s Law: Mean number in system = Arrival rate X Mean time spent in the system
Processes: Markov  Memoryless, Birth-death  Adjacent states Poisson  IID and exponential inter-arrival

32. Analysis of A Single Queue

33. Overview Birth Death Processes M/M/1 Queue M/M/m Queue
M/M/m/B Queue with Finite Buffers
Results for other Queueing systems

34. Birth-Death Processes
Jobs arrive one at a time (and not as a batch)
State = Number of jobs n in the system
Arrival of a new job changes the state to n+1  birth
Departure of a job changes the system state to n-1  death
State-transition diagram: 1 2
j-1 j
j+1 … l0 l1 l2
lj-1 lj
lj+1 m1 m2 m3 mj
j+1
 j+2

35. Birth-Death Processes (cont’d)
When the system is in state n, it has n jobs in it.
The new arrivals take place at a rate ln
The service rate is mn
We assume that both the inter-arrival times and service times are exponentially distributed

36. Theorem: State Probability
The steady-state probability pn of a birth-death process being in state n is given by:
Here, p0 is the probability of being in the zero state

37. Proof
Suppose the system is in state j at time t. There are j jobs in the system. In the next time interval of a very small duration Dt, the system can move to state j-1 or j+1 with the following probabilities:

38. Proof (cont’d)
If there are no arrivals or departures, the system will stay in state j and, thus:
Dt = small  zero probability of two events (two arrivals, two departure, or a arrival and a departure) occurring during this interval
pj(t) = probability of being in state j at time t

39. Proof (cont’d) The jth equation above can be written as follows:
Under steady state, pj(t) approaches a fixed value pj, that is:

40. Proof (cont’d) Substituting these in the jth equation, we get:
The solution to this set of equations is:

41. Proof (cont’d)
The sum of all probabilities must be equal to one:

42. M/M/1 Queue
M/M/1 queue is the most commonly used type of queue
Used to model single processor systems or to model individual devices in a computer system
Assumes that the interarrival times and the service times are exponentially distributed and there is only one server
No buffer or population size limitations and the service discipline is FCFS
Need to know only the mean arrival rate l and the mean service rate m
State = number of jobs in the system 1 2
J-1 J
J+1 … l m 

43. Results for M/M/1 Queue Birth-death processes with
Probability of n jobs in the system:

44. Results for M/M/1 Queue (cont’d)
The quantity l/m is called traffic intensity and is usually denoted by symbol r. Thus:
n is geometrically distributed. Utilization of the server = Probability of having one or more jobs in the system:

45. Results for M/M/1 Queue (cont’d)
Mean number of jobs in the system:
Variance of the number of jobs in the system:

46. Results for M/M/1 Queue (cont’d)
Probability of n or more jobs in the system:
Mean response time (using the Little's law): Mean number in the system = Arrival rate × Mean response time That is:

47. Results for M/M/1 Queue (cont’d)
Cumulative distribution function of the response time:
The response time is exponentially distributed.  q-percentile of the response time

48. Results for M/M/1 Queue (cont’d)
Cumulative distribution function of the waiting time:
This is a truncated exponential distribution. Its q-percentile is given by:
The above formula applies only if q is greater than 100(1-r). All lower percentiles are zero.

49. Results for M/M/1 Queue (cont’d)
Mean number of jobs in the queue:
Idle  there are no jobs in the system
The time interval between two successive idle intervals
All results for M/M/1 queues including some for the busy period are summarized in Box 31.1 in the book.

50. Example 31.2
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 two milliseconds to forward them. Using an M/M/1 model, analyze the gateway. What is the probability of buffer overflow if the gateway had only 13 buffers? How many buffers do we need to keep packet loss below one packet per million?
Arrival rate l = 125 pps
Service rate m = 1/.002 = 500 pps
Gateway Utilization r = l/m = 0.25
Probability of n packets in the gateway = (1-r)rn = 0.75(0.25)n)

51. Example 31.2 (cont’d)
Mean Number of packets in the gateway = (r/(1-r) = 0.25/0.75 = 0.33)
Mean time spent in the gateway = ((1/m)/(1-r)= (1/500)/(1-0.25) = 2.66 milliseconds
Probability of buffer overflow
To limit the probability of loss to less than 10-6: We need about ten buffers.

52. Example 31.2 (cont’d)
The last two results about buffer overflow are approximate. Strictly speaking, the gateway should actually be modeled as a finite buffer M/M/1/B queue. However, since the utilization is low and the number of buffers is far above the mean queue length, the results obtained are a close approximation.

53. Summary
Birth-death processes: Compute probability of having n jobs in the system
M/M/1 Queue: Load-independent => Arrivals and service do not depend upon the number in the system ln= l, mn=m
Traffic Intensity: r = l/ m
Mean Number of Jobs in the system = r/(1-r)
Mean Response Time = (1/m)/(1-r)

54. M/M/1 Queue Parameters: Traffic intensity: Stability Condition:
Traffic intensity r must be less than one
Probability of zero jobs in the system: p0 = 1- r
Probability of n jobs in the system:
Mean number of jobs in the system:

55. M/M/1 Queue (cont’d) Variance of number of jobs in the system:
Probability of k jobs in the queue =

56. M/M/1 Queue (cont’d) Mean number of jobs in the queue:
Variance of number of jobs in the queue:
Cumulative distribution function of the response time:
Mean response time:
Variance of the response time:
q-percentile of the response time = E[r] ln[100/(100-q)]
90-percentile of the response time = 2.3E[r]

57. M/M/1 Queue (cont’d) Cumulative distribution function of waiting time:
Mean waiting time:
Variance of the waiting time:
q-percentile of the waiting time =
90-percentile of the waiting time =
Probability of finding n or more jobs in the system = rn

58. M/M/1 Queue (cont’d)
Probability of serving n jobs in one busy period =
Mean number of jobs served in one busy period = [Kleinrock, p. 218, eq , with ]
Variance of number of jobs served in one busy period =
Mean busy period duration =
Variance of the busy period =

59. M/M/m Queue

60. Homework #7 Due: Monday, April 14, 2008, 12:45 PM
Submit answers to exercises 31.2; 31.3; 31.4
Due: Monday, April 14, 2008, 12:45 PM
Submit a hard copy to instructor