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Network Design and Performance Analysis

Network Design and Performance Analysis. Wang Wenjie Wangwj@gucas.ac.cn. 排队论基础 The Queueing Paradigm Single Queueing Systems ( I ). The Queueing Paradigm. Paradigms are fundamental models that abstract out the essential feature of the system being studied. Here: The paradigm of the queue

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Network Design and Performance Analysis

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  1. Network Design and Performance Analysis Wang Wenjie Wangwj@gucas.ac.cn

  2. 排队论基础 • The Queueing Paradigm • Single Queueing Systems ( I )

  3. The Queueing Paradigm • Paradigms are fundamental models that abstract out the essential feature of the system being studied. Here: • The paradigm of the queue • The paradigm of the Petri Net • Queueing theory is about the statistical prediction. Itprovides a mathematical basis for understanding and predicting the behavior of communication networks.

  4. The types of prediction making(1) • How many terminals can one connect to a time sharing computer and still maintain a reasonable response time? • What percentage of calls will be blocked on the outgoing lines of a small business’s telephone system? What improvement will result if extra line are added?

  5. The types of prediction making(2) • What improvement in efficiency can one expect in adding a second processor to a computer system? Would it be better to spend the money on a second hard disc? • Web Access： Clients /Server

  6. Queueing Theory(1) • Used for analyzing network performance and delay • times to obtain service • In data networks (packet), events are random • Random packet arrivals • Random packet lengths • Physical layer  we are concerned with bit-error-rate • At the data link/network layers we care about delays • The average delay per customer (waiting in the queue +service time) • The average number of customers in the system (waiting in the queue or undergoing service)

  7. Queueing Theory(2) • In Circuit switched networks ( e.g., telephone networks), we are interested more in call blocking probability • Number of resources ( circuits) needed to limit the blocking probability ( e.g., maintaining blocking rate under 1%) • All the above quantities will be estimated in terms of known information such as: • The customer arrival rate • The customer service rate • Notes: in many cases the customer arrival and service rates are not sufficient to determine the delay characteristics of the system • More detailed ( statistical) information about the customer interarrival and service times are needed

  8. What will we be studying? # OF USERS USAGE PATTERN SERVICE CHARACTERISTICS AMOUNT of RESOURCE PERFORMANCE - waiting time - blocking - losses

  9. ELEMENTS OF A QUEING SYSTEM • The simplest system: only involves a single queue

  10. Model (1/2) • Customers from some population arrive at the system at random arrival times • is the customer arrival rate • Queuing system has identical servers • The jth customer seeks a service that will require sj units of service time from one server • If all servers are busy, arriving customer joins a queue until a server is available

  11. Model (2/2) • Service discipline specifies the order in which customers are selected from the queue – ex : FIFO, LIFO, priority, fair queuing, … • Waiting time Qj is the time jth customer is made to wait between entering the system and entering service • Total delay in the system j = Qj + sj • n = number of customers in the system (a r.v.) • nq= number of customers in the queue (a r.v.)

  12. Notation: a/b/m/K • a = type of arrival process – M (Markov) denotes Poisson arrivals, so interarrival times are exponential random variables • b = service time distribution – M (Markov) denotes exponentially-distributed – D (Deterministic) denotes constant service times – G (General) denotes service times following some general distribution • m = number of servers • K = maximum # of customers allowed in the system

  13. Networks of Queues • An “open” queueing network accepts and loses customers from/to outside world • The total number of customers in an open network varies with time • A “closed” network does not connect with the outside world and has a constant number of customers circulating throughout it.

  14. The Poisson Process(1) P[exactly one customer arriving within the interval [t, t+  t]] =  t P[no arrivals within the interval [t, t+  t]] = 1 -  t As we let t approach 0, we have a Poisson process

  15. The Poisson Process(2) Poisson Process Poisson Process Poisson Process Aggregate Process Poisson Process Poisson Process Random Split

  16. The Poisson Process(3) • Poisson Process may not always valid • If there are correlations between an individual users calls. the user has made one call ,he then wants to make a second • If there is a heavily loaded exchange. calls are blocked ,then the users will repeated attempt to go through

  17. Foundation of The Poisson Process(1) Let • Pn(t) = P( # of arrivals by time t = n ) • pij(t): probability of going from i arrivals to j arrivals in a time interval of t seconds. • the number of arrivals is the “state” of the systems •  is the transition rate associated with each transition 3 1 4 2 0     

  18. Foundation of The Poisson Process(2) The Poisson Distribution

  19. Example 1 • Noise impulses occur on a telephone line according to a Poisson process of rate  . What is the probability that no impulses occur during the transmission of a message that is t seconds long?

  20. Number of Arrivals - Mean and Variance Let be the mean number of arrivals in an interval of length t .

  21. Inter-arrival Times • The times between successive events in an arrival process are called the inter-arrival times • Let T = time between successive arrivals in a Poisson process • T is a random variable • For the Poisson process, interarrival times are exponentially distributed random variables • Distribution and density functions for an exponentially distributed random variable …

  22. Memoryless Property(1) • The exponential distribution is memoryless • What occurs after time t is independent of what occurred prior to time t • Knowledge of the past is no help in predicting the future

  23. Memoryless Property(2) • For service times: • The additional time needed to complete a customer’s service in progress is independent of when the service started • For interarrival times: • The time for the next arrival is independent ofwhen the last arrival occurred • The exponential r.v. is the only continuous r.v. that has the memoryless property • The discrete distribution with the memoryless property is the geometric distribution

  24. Example 2 • Trains arrive at a station according to a Poisson process with mean arrival rate of 1 train each 20 minutes. You get to the station and are told the last train arrived 19 minutes ago. What is the expected time until the next train arrives?

  25. Exponential Service Times • The pdf with parameter  is given: • The mean and variance are:

  26. Exponential Service Times Is the Poisson process appropriate? • Poisson process is generally a good model. if there are a large number of users (packet sources) such that • The users are similar • The users are independent

  27. Transmission Line(1)

  28. Transmission Line(2)

  29. Transmission Line(3)

  30. Network of Transmission Line(1)

  31. Network of Transmission Line(2)

  32. Window-Based Flow Control(1)

  33. Window-Based Flow Control(2)

  34. LITTLE’s FORMULA(1) The expected number in the system (queue) is equal to the arrival rate times the expected time in the system (queue)

  35. LITTLE’s FORMULA(2)

  36. LITTLE’s FORMULA(3)

  37. LITTLE’s FORMULA(4)

  38. Example 3 • Over the course of a day, customers enter a store at an average rate of 32 customers per hour. The average customer spends 12 minutes inside. How many customers do we expect to find in the store at any given time?

  39. Applications of Little’s Formula • nq(t) = # of customers waiting in the queue for the server to become available • Q = waiting time in the queue • ns(t) = # of customers being serviced • s = service time  单服务器系统利用率

  40. M/M/1 QUEUE • Single-server system • Arrivals according to Poisson process of rate  • Inter-arrival times exponential r.v. ‘s • Service times exponential r.v. ‘s mean 1/ • Infinite buffers • The queuing discipline is first-come-first-serve (FCFS).

  41. System State • Due to the memoryless property of the exponential distribution, the entire state of the system, as far as the concern of probabilistic analysis, can be summarized by the number of customers in the system, i. – the past/history (how we get here) does not matter • When a customer arrives or departs, the system moves to an adjacent state (either i+1 or i-1).

  42. Birth-Death Process • For a state transition • If the state as being a population size, it may increase by one member at a time(“birth”) • Or, it may decrease by one member at a time(“death”)

  43. Global Balance(1) • Total rate of transitions out of state n equals total rate of transitions into state n (dynamic equilibrium)

  44. Global Balance(2)

  45. Local Balance(1) • For some Markov chains, the rate of transitions from state A to state B is equal to the rate of transition from B to A

  46. Local Balance(2)

  47. Local Balance(3)

  48. Local Balance(4)

  49. Local Balance(5)

  50. Results (1/3)

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