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1. Introduction

1. Introduction. 1.1 Description of the Queueing Problem 1.2 Characteristics of Queueing Processes 1.3 Notation 1.4 Measuring System Performance 1.5 Some General Results 1.6 Simple Data Bookkeeping for Queues 1.7 Poisson Process and the Exponential Distribution

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1. Introduction

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  1. 1. Introduction 1.1 Description of the Queueing Problem 1.2 Characteristics of Queueing Processes 1.3 Notation 1.4 Measuring System Performance 1.5 Some General Results 1.6 Simple Data Bookkeeping for Queues 1.7 Poisson Process and the Exponential Distribution 1.8 Markovian Property of the Exponential Distribution 1.9 Stochastic Processes and Markov Chains 1.10 Steady-State Birth-Death Processes

  2. Customers arriving Served customers leaving Service Facility … … Discouraged customers leaving 1.1 Description of the Queueing Problem • A Typical Queueing Process Customer (not necessary a human customer) Service Facility Arriving Process Waiting for Service Leaving the System

  3. Queue Size Queue Discipline Departure Pattern … S1 S2 Sn Arrival Pattern Service Pattern n-th Stage 1st Stage 2nd Stage 1.2 Characteristics of Queueing Processes • Arrival Pattern of Customers • Service Pattern of Servers • Queue Discipline (FCFS, LCFS,RSS,PR) • System Capacity (Queue Size/Buffer Size) • Number of Service Channels • Number of Service Stages

  4. Inter-arrival Time X1 X2 X3 X4 XN time T 1.2 Characteristics of Queueing Processes • Arrival Pattern of Customers Input of a Queueing System Random Process in terms of Mean Arrival Rate or Mean Inter-arrival Time Deterministic Inter-arrival Time Bulk or Batch Arrival Impatient Customers: Balked, Reneged, Jockey Stationary Arrival Process: Arrival Pattern not Changes with Time

  5. 1.2 Characteristics of Queueing Processes • Service Pattern of Servers In service only when the queue is not empty Random Process in terms of Mean Service Rate or Mean Service Time Deterministic Service Time Single or Batch Service State Dependent Service Stationary or Non-stationary Arrivals and Services are generally assumed Mutually Independent

  6. 1.2 Characteristics of Queueing Processes • Queue Discipline FCFS (First Come, First Served) LCFS (Last Come, First Served) RSS (Random Selection for Service in) Priority (PR):higher priority customer will be served first Preemptive: resumed or start anew Non-preemptive The same priority customers served in FCFS,… Preemptive and Non-preemptive can be used simultaneously General Discipline (GD)

  7. S1 Sc S2 S1 S2 Sc Multi-channel with Single Queue A Queue for Each Channel 1.2 Characteristics of Queueing Processes • System Capacity Queue Size (Buffer Size) + No. of Servers Finite (K) or Infinite () Number of Service Channels

  8. … S1 S2 Sn n-th Stage 1st Stage 2nd Stage Feedback 1.2 Characteristics of Queueing Processes • Stages of Services • Physical Examination procedure • Manufacturing process • Routing in Telecommunications network

  9. 1.3 Notation • Notation (A/B/X/Y/Z) A: Arrival Pattern B: Service Pattern X: No. of Servers Y: System Capacity Z: Queue Discipline Example: M/M/1//FCFS  M/M/1

  10. 1.3 Notation • Queueing Notation

  11. 1.4 Measuring System Performance • Measures of Effectiveness Waiting Time (Queue waiting time, System time) Queue Length Manner Idle Time or Busy Time of Servers Usually Calculate Probability Distribution or Expected Values for Random Variables Use the Measures to design an Optimal System Applications Telephone Conversations Landing of Aircraft Telecom/Computer Networks Automatic Control Manufacturing System

  12. 1.5 Some General Results • General Results for G/G/c Arrival rate： Service rate： Traffic congestion for c-servers：=  /c 1. When  > 1,the system is unstable. 2. When  = 1,unless arrivals and service are deterministic and perfectly scheduled, no steady state exists. 3.When  < 1,the system is stable. Steady state probability for system size：pn

  13. 1.5 Some General Results • Little’s Formulas (early 1960s) • Verification of Little’s Formulas by Example

  14. 1.5 Some General Results • Verification of Little’s Formulas (Cont’d) Similar results can be obtained for Lq = Wq .

  15. 1.5 Some General Results • Summary of General Results for G/G/c Queues  = /cTraffic intensity; offered work load to a server L = W Little’s Formula Lq = Wq Little’s Formula W = Wq+1/ Expected-value argument pb = /c = Busy probability for an arbitrary server r = / Expected no. of customers in service L = Lq+r Combined result p0 = 1G/G/1 empty-system probability L = Lq+(1 p0) Combined result for G/G/1

  16. 1.6 Simple Data Bookkeeping For Queues • Deterministic Queue D/D/1/K-1 Constant arrival rate Constant service rate  The system is empty at time t = 0 Whenever a customer to the system would increase the total no. in the system to be K, he is refused entry. As soon as a service is completed another is begun. FCFS queueing discipline is employed  n(t) : the no. in the system at time t  : the waiting time of the n-th arrival How to find n(t) and ?

  17. a = 3,  = 1/3,  = 1/5 arrival a 0 t time departure D/D/1/K-1 (under  > ) • How to find n(t)?

  18. arrival a 0 t time departure D/D/1/K-1 (under  > ) • How to find n(t)?

  19. D/D/1/K-1 (under  > ) • How to find n(t)? Similarly,

  20. D/D/1/K-1 (under  > ) • How to find n(t)?

  21. D/D/1/K-1 (under  > ) • How to find n(t)?

  22. How to find ? n-th arrival (n+1)-st arrival time (n-1)-st departure n-th departure D/D/1/K-1 (under  > )

  23. How to find ? (Assume ) D/D/1/K-1 (under  > )

  24. arrival 0 time departure D/D/1/K-1 (under  > ) • Example

  25. arrival 0 time departure D/D/1/K-1 (under  > ) • Example (cont’d)

  26. How to find n(t) and ? arrival a 0 t time departure D/D/1/K-1 (under  < )

  27. n(t) and with M customers at start M arrival 0 t time departure D/D/1/K-1 (under  < )

  28. n(t) and with M customers at start M arrival 0 t time departure D/D/1/K-1 (under  < )

  29. i Interarrival time between customers i+1 and i Service time of customer i Event-Oriented Bookkeeping • Input Data

  30. No. in Queue Just After Master Clock Time No. in System Just After Master Clock Time Event-Oriented Bookkeeping

  31. 1.7 Poisson Process and The Exponential Distribution • Counting Process

  32. Poisson Process • Independent Increments A counting process is said to posses independent increments if the numbers of events that occur in disjoint time intervals are independent. • Poisson Process: Definition 1

  33. independent increment stationary Poisson Process • Property of Poisson Process by Definition 1

  34. Poisson Process • Property of Poisson Process (cont.)

  35. Poisson Process • Property of Poisson Process (cont.)

  36. Poisson Process • Poisson Process: Definition 2 Definitions 1 & 2 for Poisson Process are equivalent.

  37. Poisson Process • Properties of Poisson Process 1. Poisson process has stationary and independent increments. 2. The interarrival timeT of Poisson Process follows the exponential distribution. That is, the cumulative function of T , A(t) satisfies The probability density function a(t) is given by

  38. Poisson Process • Proof of Property 2 It can also be shown that if the interarrival times are independentand have the same exponential distribution, then the arrival rate follows the Poisson distribution.

  39. Poisson Process Pf : The sum of independent and identical exponential random variables have an Erlang distribution,

  40. Poisson Process

  41. Poisson Process • Property of Poisson Process 3. The mean arrival rate is , and the mean interarrival time is 1/. 4. Given the N(T) = k, the k arrival times 1, 2, …, k have the same distribution as the order statistics corresponding to k independent random variables uniformly distributed on the interval (0, T].

  42. Poisson Process • Proof of Property 4 of Poisson Process

  43. 1.8 Markovian Property of the Exponential Distribution • What is Markovian Property? • Memorylessness property of the exponential distribution: the probability that a customer in service will be completed at some future time t is independent of how long he has already been in service Pf:

  44. Markovian Property • Memorylessness Property (cont.) • Memorylessness implies the exponential distribution: Pf :

  45. Generalization of Markov Process • Unconstant Arrival Rate • Compound Poisson Process (Batch Arrival)

  46. 1.9 Stochastic Process and Markov Chains • Stochastic/Random Process A family of Random Variables, X(t) Discrete-parameter Stochastic Process Continuous-parameter Stochastic Process

  47. 1.9.1 Markov Process • Definition: xi is called a state Given the “present” condition, the “future” is independent of the “past”, and the process is thus “memoryless.”

  48. 1.9.1 Markov Process • Classification of Markov Process • Markov Chains The sequence {Xn} there are only finite or countably infinite number of states.

  49. 1.9.2 Discrete-Time Markov Chains • Discrete-Parameter Markov Chains The system is said to be in state j after n steps or transitions. The conditional probability Pr{Xn= j|Xn-1= i} is called the single-step transition probability or just transition probability. If the transition probabilities are independent of n, then the chain is said to be homogeneous and denoted as pij .

  50. i j pji Bayes Theorem pij 1.9.2 Discrete-Time Markov Chains • Notations of Discrete-Parameter Markov Chains • Chapman-Kolmogorov (C-K) Equations Pf :

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