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Network Design and Analysis. Wang Wenjie Wangwj@gucas.ac.cn. Queueing System IV Discrete-Time Queuing Systems. 1. Discrete-Time Queuing Systems. 1. 1. Motivation 1. 2. The Bernoulli Process 1. 3. Geo/Geo/1 Systems 4. Geo/Geo/1/N Systems 1. 5. Simple ATM Queuing Systems.
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Network Design and Analysis Wang Wenjie Wangwj@gucas.ac.cn
Queueing System IV Discrete-Time Queuing Systems
1. Discrete-Time Queuing Systems 1. 1. Motivation 1. 2. The Bernoulli Process 1. 3. Geo/Geo/1 Systems • 4. Geo/Geo/1/N Systems 1. 5. Simple ATM Queuing Systems
1.1 Motivation Some queuing systems operate on a slotted time basis: DT Systems
Model • All events (arrivals, departures) must occur at integer multiples of the slot time TS • Implication: all service times must be multiples of TS • For convenience, often use normalized time • Example: ATM – Fixed-size cells imply all service times equal 1 time slot (on a given link)
Discrete-Time Markov Chain • Let j(n) = P[ X(n) = j ], j= 0, 1, … • This denotes the probability of being in state j at time n , so 1- 1- Off On
Properties of P • A square matrix whose dimension is the size of the state space • Every row adds up to 1 (n+m) = (n) Pm • Therefore, if we know the initial state pmf (0), we can get the state pmf at any time m. • If DTMC is stationary, in steady-state: = P
Exercise 1.1 An urn initially contains 5 black balls and 5 white balls. The following experiment is repeated indefinitely: • A ball is drawn from the urn • If the ball is white it is put back in the urn • If the ball is black it is left out X(n) is the number of black balls after n draws from the urn
Exercise 1.1(Cont’d) • Draw the Markov chain and find the transition probabilities. • Find the matrix P. • Find the probability that there are 4 black balls in the urn after 2 draws.
1.2. The Bernoulli Process • Discrete-time analogous to Poisson process – P[1 arrival during a time slot] = a – P[0 arrivals during a time slot] = 1-a • Mean # of arrivals during a time slot = ____ • Motivation: synchronous high-speed packet switches where at most one packet can be transmitted over a link during a slot • Similarities to the Poisson process • Memoryless • Multiplexing and demultiplexing of a Bernoulli process still result in Bernoulli processes.
Geometric Distribution (1/2) • P[next arrival occurs within k time slots] = P[k-1 empty slots] P[arrival] = (1-p)k-1 p Geometric distribution • Poisson process Exponential Interarrival Time • Bernoulli process Geometric Interarrival Time
Geometric Distribution (2/2) • Geometric distribution has the memoryless property • Mean : 1/p • Variance : (1-p)/p2
Binomial Distribution • N slots with i arrivals in some sequence : P(sequence)=pi(1-p)N-i • There are i arrivals in N slots for some value of p, the binomial distribution is : b(i ,N, p)= CiNpi(1-p)N-i • Mean : Np
1.3. Geo/Geo/1 Systems • Bernoulli arrivals, geometric service times (DT analog to M/M/1)
Model • Let a = P[arrival] • Let s = P[service completion] (departure) Note that this is a discrete time Markov chain, so these are transition probabilities .NOT transition rates.
Local Balance Equations Analogous to in M/M/1 system
Average # of Customers If 0< r <1 then:
Mean Delay • System throughput (customers/slot time) = a = • Apply Little’s law
1.4. Geo/Geo/1/N Systems • Similar to Geo/Geo/1, except for state N
Local Balance Equations • As before, local balance yields • However Now:
Delay • P[block] = P[N in system] = N • So, throughput is a(1- N)
Simple ATM Queuing System(2) • Deterministic service time of one time slot • Normalized to cell transmission time • Always a service completion if non-empty • Server is never idle if a job (cell) is waiting • Simple Bernoulli is not interesting since there is no queuing • Of interest are systems where there can be up to M > 1 arrivals per time slot, e.g., from M different input sources
M-Geo/D=1/1/N Systems (3) Transition Matrix
