TELCOM 2130 Queueing Theory David Tipper Associate Professor Graduate Telecommunications and Networking Program University of Pittsburgh Learning Objective To develop the modeling and mathematical skills to analytically determine computer systems and communication network performance. Students should be able to read and understand the current performance analysis and queueing theory literature upon completion of the course. Understand strengths and weaknesses of Queueing Models 2 1
Texts and Reading Assignments System Modeling and Analysis: Foundations of System Performance Evaluation, H. Kobayashi and B. Mark Similar book is Performance Analysis of Communications Networks and Systems, Piet Van Mieghem Classic Texts are Queueing Systems, Theory, Volumes I and II, L. Kleinrock, 1976 Fundamentals of Queueing Theory, 4 th edition, D. Gross, J. Shortle, J. Thompson, and C. Harris, 2008. Free Reference ITU-D Teletraffic Handbook For Background suggest Probability, Statistics, and Random Processes For Electrical Engineering (3rd Edition) by Alberto Leon-Garcia (Jan 7, 2008) Matrices: Methods and Applications, Stephen Barnett, 1990. Course Outline Course Organization, Reading : KM 1 Review of Background Material Reading : ALG, SB, KM 5.1 Birth-Death Queues Reading: KM 2 Loss Models Reading : KM 3 Non-Markovian Queues Reading : KM 4, 5 Queueing Networks Reading : KM 6-8 Phase Type Queues Reading : KM 10 Discrete Time Queues Reading : KM 11 Time Dependent Solution of Queues Reading : KM 15 2
Grading Grading Homework 25% 6-7Assignments Midterm Exam 20% Final exam 25% Term project 30% 5-8 pages IEEE Two Column Paper Format List of possible projects will be provided or topic can be proposed by students in mutual agreement with Professor Individual project based around research papers in the literature Past projects Backbone Buffer Sizing Performance analysis of hybrid optical switching Markov models of WLAN VoIP capacity Survey of Queues with Negative Customers 5 Queueing Theory Queueing theory : Mathematical analysis of waiting lines Queueing Theory is the primary tool for evaluating performance in the initial stage of system design. Analytical Model of the system based on stochastic process and probability theory Approximates real system by focusing on contention at shared resources. Examples: shared medium router, window flow controlled session, time shared computer system 6 3
History of Queueing Orginated with A.K. Erlang in 1917, Dutch mathematician - hired by Copenhagen Phone company Determine how many operators to keep call blocking at reasonable level Results still used today Queueing Theory often called Teletraffic Theory until the 1950 s when applications to service systems, manufacturing systems and transportation systems developed. L. Kleinrock in dissertation at MIT first to apply queueing theory to data networks in early 60 s basic packet switching and delay analysis Sauer, Chandy applied queueing theory to computer systems in 60 s Studied how disk drives interact with computer processing power, time sharing strategies 7 Model of Router 8 4
Model of Time Shared Computer Systems 9 Flow Controlled Session Router A Router B Source 2 3 Data Data Data Destination 1 Router C Router D 4 ACK 6 5 ACK ACK 5
Nomenclature of a Queueing System The input process how customers arrive The system structure waiting space number of servers, etc. The service process Kendall s Notation 1/2/3/4/5/6 A Shorthand notation to describe a queueing system containing a queueing system. 1 : Customer arrival pattern (Interarrival times distribution). 2 : Service pattern (Service-times distribution). 3 : Number of parallel servers. 4 : System capacity. 5 : Queueing discipline. 6: Customer Population 11 Characteristics of the Input Process (1) Arrival pattern or Arrival Process Customers may arrive at a queueing system either in some regular pattern or in a random fashion. When customers arrive regularly at a fixed interval, the arrival pattern can be easily described by a single number the rate of arrival When customers arrive according to some random fashion, the arrival pattern is described by a probability distribution. Arrival process characterized by interarrival distribution 12 6
Characteristics of the Input Process (2) Probability distribution that are commonly used to describe the interarrival process are: M : Markovian (or memoryless) same as exponentially distributed interarrivals - same as Poisson arrival process (number of customers arrive over an interval has a Poisson distribution) D : Deterministic, fixed interarrival times E k : Erlang distribution of order k PH phase type distribution Geo Geometric distribution G : General probability distribution GI: General and independent (inter-arrival time) distribution. 13 Characteristics of the Input Process (3) Behavior of the arriving customers Customer arriving at a queueing system may behave differently when the system is full (due to finite waiting queue) or when all servers are busy. Blocking System : The arriving customers when system is full are considered lost dropped from systems Non-Blocking System : The arriving customers are placed in queues of infinite size. Balking or Discouraged arrivals : customers refuse to join queue when line too long or the arrival rate decreases with line length 14 7
Characteristics of the Service Process Service distribution describe the time take by a server to process a customer. Can be deterministic or probabilitistic In fashion similar to interarrival process use abbreviations to describe common cases M : Markovian (or memoryless), implies the exponentially distributed service times. D : Deterministic, constant service times E k : Erlang distribution of order k service time distribution PH phase type distribution G : General service time distribution 15 Characteristics of the System Structure (1) Physical number and layout of servers Default assumption of parallel and identical servers Integer number of serviers A customer at the head of the waiting queue can go to any server who is free, and leave the system after receiving service from that server. The system capacity The system capacity is the maximum number of customers that a queueing system can accommodate, inclusive of those customers at the service facility Finite (integer value) - maximum number of customers in the systems Infinite (default value) 16 8
Characteristics of the Service Process Queueing discipline how customers are selected for service from the line First-Come-First-Served (FCFS/FIFO) Last-Come-First-Served (LCFS) Priority Process sharing Random Longest Queue First Etc. The size of the customer population Infinite : the number of potential customers from external sources is very large as compared to those in the system. Finite : the arrival process (rate) is affected by the number of customers already in the system. 17 Example Consider a link modeled as the superposition of S traffic streams as a multiple Poisson streams each with mean rate i,, and exponentially distributed service times with mean rate C, one server, infinite waiting space, first in first out service and an infinite population of customers M 1 + M 2 + M S / C/1/inf/FIFO/inf 18 9
Example of Notation M/D/2/50/FIFO/ 1. Exponentially distributed interarrival times 2. Deterministic service times 3. Two parallel servers 4. Waiting space for 48 customers + 2 in service 5. First in First Out processing from the queue 6. Infinite population of customers 19 Nomenclature Standard notation mean arrival rate of customers/time unit mean service rate in customers/time unit n(t) number of customers in the system at time t π i = lim t P{n(t) = i} is server utilization remember for stability L Average number of customers in systems L q - Average number of customers in the queues know L = L q + W Average delay in system (includes server + queue) W q Average delay in queue know W = W q + 1/ Little s Law L = W 10
Nomenclature Standard notation - relationships Mathematical Background for Queueing Queueing Theory draws on results from Probability Theory Discrete and continuous random variables PDF, CDF, etc. Joint random variables Moments and functions of RVs, etc. Transform Techniques Z transform, Laplace transform, Fourier transform, etc. Moment generating functions Linear Algebra - matrix based methods and analysis Matrix Inversion, Eigenvalues/eigenvectors spectral decomposition, Kronecker Products 22 11
Summary Overview of Course Queueing Theory and Performance Modeling Notation and Nomenclature 23 12