1 Performance Evaluation of Computer Networks Objectives  Introduction to Queuing Theory  Little’s Theorem  Standard Notation of Queuing Systems  Poisson Process and its Properties  M/M/1, M/M/m, M/M/m/m, and M/G/1 Queuing System  Network of queues  Jackson Networks

2 Introduction  Each one of us has spent a great deal of time waiting in lines.  One example in the Cafeteria  Other examples of queues are  Printer queue  Packets arriving to a buffer  Calls waiting for answer by a technical support

3 What makes up a queue?  The System: A collection of objects under study  It is important to define the system boundaries  The Entities: The people, packets, or objects that enter the system requiring some kind of service  The Servers: The people, resources, or servers that perform the service required  The Queue: An accumulation of entities that have entered the system but have not been served

4 Queue Discipline  First Come First Served - FCFS  Most customer queues  Last Come First Served - LCFS  Packages, Elevator  Served in Random Order - SIRO  Entering Buses  Priority Service  Multi-processing on a computer  Emergency room

5 What factors effect system performance TThe Arrivals Process TThe time between any two successive arrivals DDoes this depend on the number of packets in the system? FFinite populations TThe Service Process TThe time taken to perform the service DDoes this depend on the number of packets in the system? TThe number of servers operating in system TThe Service Discipline SSystem Capacity PProcesses waiting + processes being served

6 Measuring System Performance TThe total time an “entity” spends in the system (Denoted by W) TThe time an “entity spends in the queue ( Denoted by W q ) TThe number of “entities” in the system ( Denoted by L ) TThe number of “entities” in the queue ( Denoted by L q ) TThe percentage of time the servers are busy (Utilization time) These quantities are variable over time

7 What is Queuing Theory?  Primary methodological framework for analyzing network delay  Often requires simplifying assumptions since realistic assumptions make meaningful analysis extremely difficult  Provide a basis for adequate delay approximation queue

8 Packet Delay  Packet delay is the sum of delays on each subnet link traversed by the packet  Link delay consists of:  Processing delay  Queuing delay  Transmission delay  Propagation delay node packet delay link delay

9 Link Delay Components (1)  Processing delay  Delay between the time the packet is correctly received at the head node of the link and the time the packet is assigned to an outgoing link queue for transmission head nodetail node outgoing link queue processing delay

10 Link Delay Components (2)  Queuing delay  Delay between the time the packet is assigned to a queue for transmission and the time it starts being transmitted head nodetail node outgoing link queue queuing delay

11 Link Delay Components (3)  Transmission delay  Delay between the times that the first and last bits of the packet are transmitted head nodetail node outgoing link queue transmission delay

12 Link Delay Components (4)  Propagation delay  Delay between the time the last bit is transmitted at the head node of the link and the time the last bit is received at the tail node head nodetail node outgoing link queue propagation delay

13 Queuing System (1)  Customers (= packets) arrive at random times to obtain service  Service time (= transmission delay) is L/C  L : Packet length in bits  C : Link transmission capacity in bits/sec queue customer (= packet) service (= packet transmission)

14 Queuing System (2)  Assume that we already know:  Customer arrival rate  Customer service rate  We want to know:  Average number of customers in the system  Average delay per customer customer arrival rate customer service rate average delay average # of customers

15 Little’s Theorem

16 Definition of Symbols (1)  p n = Steady-state probability of having n customers in the system  = Arrival rate (inverse of average interarrival time)   = Service rate (inverse of average service time)  N = Average number of customers in the system

17 Definition of Symbols (2)  N Q = Average number of customers waiting in queue  T = Average customer time in the system  W Q = Average customer waiting time in queue (does not include service time)

18 Little’s Theorem  N = Average number of customers  = Arrival rate  T = Average customer time in the system N = T  Hold for almost every queuing system that reaches a steady-state  Express the natural idea that crowded systems ( large N ) are associated with long customer delays ( large T ) and reversely

19 Application of Little’s Theorem (2)  Consider a window flow control system  W : Window size   : Packet arrival rate  T : Average packet delay  From Little’s Theorem W >= T  If T increases, must eventually decrease  If is limited due to congestion, increasing W merely serves to increase T

20 Standard Notation of Queuing Systems

21 Standard Notation of Queuing Systems (1) X/Y/Z/K  X indicates the nature of the arrival process  M : Memoryless (= Poisson process, exponentially distributed interarrival times)  G : General distribution of interarrival times  D : Deterministic interarrival times  Y indicates the probability distribution of the service times  M : Exponential distribution of service times  G : General distribution of service times  D : Deterministic distribution of service times

22 Standard Notation of Queuing Systems (2) X/Y/Z/K  Z indicates the number of servers  K (optional) indicates the limit on the number of customers in the system  Examples:  M/M/1, M/M/m, M/M/∞, M/M/m/m  M/G/1, G/G/1  M/D/1, M/D/1/m

23 Poisson Process and its Properties

24 The Poisson Arrival Model  A Poisson process is a sequence of events “randomly spaced in time”  Examples  Customers arriving to a bank  Packets arriving to a buffer  The rate λ of a Poisson process is the average number of events per unit time (over a long time)

25 Properties of Poisson Process (1)  Interarrival times  n are independent and exponentially distributed with parameter  The mean and variance of interarrival times  n are 1/ and 1/ ^2, respectively

26 Properties of Poisson Process (2)  If two or more independent Poisson process A 1,..., A k are merged into a single process A = A 1 + A A k, the process A is Poisson with a rate equal to the sum of the rates of its components A1A1 AiAi AkAk A independent Poisson processes Poisson process merge 1 i k

27 Properties of Poisson Process (3)  If a Poisson process A is split into two other processes A 1 and A 2 by randomly assigning each arrival to A 1 or A 2, processes A 1 and A 2 are Poisson A1A1 A2A2 A Poisson processes Poisson process split randomly 1 2 with probability p with probability (1-p)

28 M/M/1 Queuing System

29 M/M/1 Queuing System  A single queue with a single server  Customers arrive according to a Poisson process with rate  The probability distribution of the service time is exponential with mean 1/  Poisson arrival with arrival rate Exponentially distributed service time with service rate  single server infinite buffer

30 M/M/1 Queuing System: Results (1)  Utilization factor (proportion of time the server is busy)  Probability of n customers in the system  Average number of customers in the system

31 M/M/1 Queuing System: Results (2)  Average customer time in the system  Average number of customers in queue  Average waiting time in queue

32 M/M/m Queuing System

33 M/M/m Queuing System  A single queue with m servers  Customers arrive according to a Poisson process with rate  The probability distribution of the service time is exponential with mean 1/  Poisson arrival with arrival rate Exponentially distributed service time with rate  m servers infinite buffer 1 m

34 M/M/m Queuing System: Results (1)  Ratio of arrival rate to maximal system service rate  Probability of n customers in the system

35 M/M/m Queuing System: Results (2)  Probability that an arriving customer has to wait in queue (m customers or more in the system)  Average waiting time in queue of a customer  Average number of customers in queue

36 M/M/m Queuing System: Results (3)  Average customer time in the system  Average number of customers in the system

37 M/M/m/m Queuing System

38 M/M/m/m Queuing System  A single queue with m servers (buffer size m)  Customers arrive according to a Poisson process with rate  The probability distribution of the service time is exponential with mean 1/  Poisson arrival with arrival rate Exponentially distributed service time with rate  m servers buffer size m 1 m

39 M/M/m/m Queuing System: Results  Probability of m customers in the system  Probability that an arriving customer is lost

40 M/G/1 Queuing System

41 M/G/1 Queuing System  A single queue with a single server  Customers arrive according to a Poisson process with rate  The mean and second moment of the service time are 1/  and X 2 Poisson arrival with arrival rate Generally distributed service time with service rate  single server infinite buffer

42 M/G/1 Queuing System: Results (1)  Utilization factor  Mean residual service time

43 M/G/1 Queuing System: Results  Pollaczek-Khinchin formula

44 Network of Queues  Network is a model in which jobs departing from one queue arrive at another queue (or possibly the same queue)  Open Networks: all customers can leave the network  Closed Networks: No customers can leave the network

45 Jackson Networks  Jackson Network is named after James R. Jackson  It is the first significant development in the theory of networks of queues  Each node of the queueing network can be analyzed separately  The utilization of all of the queues is less than one

46 Open Jackson Networks  There are J queues  Customers arrive at queue l according to independent Poisson processes with rate  The service times in queue l are exponential with rates  Upon leaving queue l, each customer is sent to queue m with probability and leaves the network with probability  The routing decision is independent of the past evolution of the network

47 An Open Jackson Network i j k m

48 Conclusion  Queuing models provide qualitative insights on the performance of computer networks, and quantitative predictions of average packet delay  To obtain tractable queuing models for computer networks, it is frequently necessary to make simplifying assumptions  A more accurate alternative is simulation, which, however, can be slow, expensive, and lacking in insight