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# notes08_Mod - 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 2 Stochastic Processes (recap) • A stochastic process is a mathematical model for describing an empirical process that changes in time according to some probabilistic forces. • A stochastic process is a family of random variables { X(t) , t T } defined on a given probability space, indexed by the time parameter t where t is in an index set T . • The probability that X(t) takes on a value, say i and that is P[ X(t)=i ], is the range of that probability space. 3 Characteristics of Stochastic Processes (recap) Basically, there are three parameters that characterize a stochastic process. State Space – The values assumed by a random variable X(t) are called “states” and the collection of all possible values forms the “state space” of the process. – If X(t)=i , then we say the process is in state i . – Discrete-state process • The state space is assumed to be the non-negative integers {0, 1, 2,…}. – Continuous-state process • The state space contains finite or infinite intervals of the real values.

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2 4 Characteristics of Stochastic Processes (recap) Index parameter – The index is always taken to be the time parameter. – Discrete-time process • A process changes state (or makes a “transition”) at discrete or finite countable time instants. – Continuous-time process • A process may change state at any instant on the time axis. 5 Characteristics of Stochastic Processes (recap) Statistical dependency – Statistical dependency of a stochastic process refers to the relationships between one random variable and other members of the same family. –F o r Markov process , its future probabilistic development is dependent only on the most current state, how the process arrives at the current position is irrelevant. 6 Characteristics of Stochastic Processes (recap) Continuous time stochastic process Continuous time stochastic chain Continuous Time Discrete time stochastic process Discrete time stochastic chain Discrete Time Continuous State Discrete State State Space Time Parameters
3 7 Discrete Time Markov Chains (1) • The stochastic sequence { X k , k T } is a Markov Chain if the following conditional probability holds for all i , j and k . P[ X k+1 = j | X 0 = i 0 , X 1 = i 1 ,…, X k-1 = i k-1 , X k = i ] = P[ X k+1 = j / X k = i ] = P i j • The future probability development of the chain depends only on its current state (kth instant) and no only on how the chain has arrived at the current state. • “Memoryless” chain. 8 Discrete Time Markov Chains (2) P i j is (one-step) transitional probability, which is the probability of the chain going from state i to state j. P i j is a function of time. If it does not vary with time (independent of k), then the chain is said to have stationary transition probability.

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notes08_Mod - Introduction to Queueing Systems TELCOM 2120...

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