Computer Architecture II: Specialized 0909.444.01/02 Fall 2001

1. Computer Architecture II: Specialized0909.444.01/02Fall 2001 Lecture 4November 7, 2001 John L. Schmalzel Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/fall01/comparch2/

2. Plan • 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. Real Number, a Random Event, s Random Variable, X Recall: Random Variable • 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 Couch Appendix B Prob & RV Real Number Random Variable Random Event

4. Cumulative Distribution Function (CDF) of x Probability Density Function (PDF) of x f(x) x a b Recall: Parameters of an RV F(a) a

5. Experiment: E1 v(t,E1) Real Number, a Real function of time, v(t) Random Event, E Random Event, s t Noise Source Random Variable, X Random Process, X v(t) t2 t2 t2 t1 t1 t1 Experiment: E2 v(t,E2) t Experiment: E3 v(t,E3) t Random/Stochastic Process v(t) = {v(t,Ei)} : Random/Stochastic Process

6. l 0 1 m 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

7. l l l 2 3 4 . . . . . . . m m m l 0 1 m Markov Chains

8. Queuing Theory for Packet Transmission Attributes of a Queue • Interarrival-time pdf • Service-time pdf • Number of servers • Queuing discipline • Buffer (waiting) space

9. Queue Server Mean arrival rate l customers/sec Mean service rate m customers/sec Single Server Queue

10. Probability of m arrivals in time t Probability of occurrence of queue length n Average delay Average queue length Average arrival rate Utilization factor 7 Number joined queue by time t Service time For 2nd customer 6 5 Queuing time For 2nd customer Length of queue between t4 and t5’ 4 Number of customers 3 Number served by time t 2 Waiting time For 2nd customer 1 Time Arrivals Departures (service) 0 t1 t2 t3 t4 t5 t6 t1’ t2’ t3’ t4’ t5’ t6’ Queue Performance

11. Queue (1-ldt) Mean arrival rate l customers/sec A A A A A A A A Pj Pj+1 Pj-1 ldt ldt Time t t dt The Arrival Process Random interarrival-times Markov model

12. Summary

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