notes07 - Introduction to Queueing Systems TELCOM 2120...

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1 Introduction to Queueing Systems TELCOM 2120 Network Performance Joseph Kabara Telecommunications Program University of Pittsburgh Spring 2008, notes7 TELCOM 2120: Network Performance 2 Queueing Theory • Queueing theory : Mathematical analysis of waiting lines • Queueing Theory is the primary analytical framework for evaluating performance in the initial stage of system design. • Analytical Model of the system – based on stochastic processes • Approximates real system by focusing on contention at shared resources. • Examples: shared medium router, window flow controlled session, time shared computer system Spring 2008, notes7 TELCOM 2120: Network Performance 3 Shared Medium ATM Switch μ 1 2 N 1 o λ 2 o o N 1 i 2 i i N Shared Medium Switch
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2 Spring 2008, notes7 TELCOM 2120: Network Performance 4 Model of Router Spring 2008, notes7 TELCOM 2120: Network Performance 5 Modeling Window Flow Control Concentrate on a single session in network Model as tandem series of queues Spring 2008, notes7 TELCOM 2120: Network Performance 6 Model of Time Shared Computer Systems Time sharing system A Queuing Network Model of a Time Sharing System
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3 Spring 2008, notes7 TELCOM 2120: Network Performance 7 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 arriving pattern (Interarrival times distribution). – 2 : Service pattern (Service-times distribution). – 3 : Number of parallel servers. – 4 : System capacity. – 5 : Queueing discipline. – 6: Customer Population Spring 2008, notes7 TELCOM 2120: Network Performance 8 Characteristics of the Input Process (1) 1. The size of the arriving 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. 2. Arriving patterns – Customers may arrive at a queueing system either in some regular patterns or totally random fashions. When customers arrive regularly at a fixed interval, the arriving pattern can be easily described by a single number the rate of arrival When customers arrive according to some random fashion, the arriving pattern is described by statistical distribution. Spring 2008, notes7 TELCOM 2120: Network Performance 9 Characteristics of the Input Process (2) • Probability distribution that are commonly used to describe the arrival process are: M : Markovian (or memoryless), imply the Poisson process. D : Deterministic, fixed interarrival times E k : Erlang distribution of order k G : General probability distribution GI : General and independent (inter-arrival time) distribution.
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4 Spring 2008, notes7 TELCOM 2120: Network Performance 10 Characteristics of the Input Process (3) 3. Behavior of the arriving customers
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notes07 - Introduction to Queueing Systems TELCOM 2120...

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