1. S. Mandayam/ CompArch2/ECE Dept./Rowan University Computer Architecture II: Specialized 0909.444.01/02 Fall 2001 John L. Schmalzel Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/fall01/comparch2/ Lecture 4 November 7, 2001

2. S. Mandayam/ CompArch2/ECE Dept./Rowan UniversityPlan Recall: Random Variables Random Processes Markov Random Process Queuing Theory for Packet Transmission Model Attributes Single server queue Arrival Process Service Process Performance Parameters

3. S. Mandayam/ CompArch2/ECE Dept./Rowan University Recall: Random Variable Random Event, s Real Number, a Random Variable, X Definition: Let E be an experiment and S be the set of all possible outcomes associated with the experiment. A function, X, assigning to every element s S, a real number, a, is called a random variable. X(s) = a Random Variable Random Event Real Number Couch Appendix B Prob & RV

4. S. Mandayam/ CompArch2/ECE Dept./Rowan University Recall: Parameters of an RV Cumulative Distribution Function (CDF) of x Probability Density Function (PDF) of x x f(x) a b a F(a)

5. S. Mandayam/ CompArch2/ECE Dept./Rowan University Random/Stochastic Process Experiment: E 1 Experiment: E 2 v(t,E 1 ) v(t,E 2 ) v(t,E 3 ) t t t t1t1 t2t2 Experiment: E 3 t1t1 t2t2 t1t1 t2t2 Noise Source v(t) v(t) = {v(t,E i )} : Random/Stochastic Process Random Event, s Real Number, a Random Variable, X Random Event, E Real function of time, v(t) Random Process, X

6. S. Mandayam/ CompArch2/ECE Dept./Rowan University Markov Chains Models of random evolution using random process theory Allows calculation of the probabilities for the change in state of a countable set Memory-less system Solutions for the predictions on the state of a Markov chain do not depend on time 1 0 

7. S. Mandayam/ CompArch2/ECE Dept./Rowan University Markov Chains 2  3  4  1 0 .......

8. S. Mandayam/ CompArch2/ECE Dept./Rowan University Queuing Theory for Packet Transmission Attributes of a Queue Interarrival-time pdf Service-time pdf Number of servers Queuing discipline Buffer (waiting) space

9. S. Mandayam/ CompArch2/ECE Dept./Rowan University Single Server Queue Mean arrival rate customers/sec Mean service rate  customers/sec Server Queue

10. S. Mandayam/ CompArch2/ECE Dept./Rowan University Queue Performance Probability of m arrivals in time t Probability of occurrence of queue length n Average delay Average queue length Average arrival rate Utilization factor 0 1 2 3 4 5 6 7 t 1 t 2 t 3 t 4 t 5 t 6 t 1 ’ t 2 ’ t 3 ’ t 4 ’ t 5 ’ t 6 ’ Number of customers Time Arrivals Departures (service) Queuing time For 2 nd customer Waiting time For 2 nd customer Service time For 2 nd customer Number joined queue by time t Number served by time t Length of queue between t 4 and t 5 ’

11. S. Mandayam/ CompArch2/ECE Dept./Rowan University The Arrival Process Mean arrival rate customers/sec Queue P j-1 tt PjPj P j+1 tt  t) Markov model Time A AAAA AA A  tt t Random interarrival-times

12. S. Mandayam/ CompArch2/ECE Dept./Rowan UniversitySummary

13. References I. A. Glover and P. M. Grant, Digital Communications, Prentice-Hall, 1998. Jean Walrand, Communication Networks, 2 nd Edition, WCB/McGraw- Hill, 1998.