1. Lecture 10: Queueing Theory

2. Queueing Analysis Jobs serviced by the system resources Jobs wait in a queue to use a busy server queueserver

3. Queueing Network Models Open Network Model Jobs arrive in the system from an external resource and depart the system Closed Network Model Has a fixed number of jobs that circulate among various queues and servers

4. Queueing Analysis saverage service time  average service rate (1/s) average arrival rate ((number of jobs arrive in time T) / T)  traffic intensity ( /  ) raverage response time (waiting time + service time) waverage waiting time in queue qaverage number of jobs in the queue n number of jobs in the system (waiting & being serviced) Uutilization (busy time / total time) anumber of arrivals within interval T dnumber of departures within interval T

5. Operational Analysis Average arrival rate = (requests/s) Utilization U = U = s  Utilization Law U = =  Traffic intensity Computer system Job arrival rate (  Job service rate ( 

6. Operational Analysis Ex: A single disk system Average time to service each request is 600  s. What is the maximum possible request rate it can tolerate?

7. Operational Analysis Utilization U = s U = =  Traffic intensity  > 1arrival rate > service rate number of jobs in the queue increase without bound  < 1arrival rate < service rate stable system

8. Operational Analysis Little’s Law D = w T D = q T  q = w Little’s Law

9. q = w U = sn = r Operational Analysis queueserver

10. Operational Analysis Ex: Average response time of a web server is 11 ms. Request rate is 450 requests/s What is the average number of requests that are buffered in the system?

11. Stochastic Analysis Kendall’s Notation A/S/c A – arrival process most common arrival process is Poisson process: interarrival times follow exponential distribution exponential distribution is memoryless S – service process Poisson service process is the most common c – number of servers

12. Stochastic Analysis Interarrival time: Time between successive jobs arriving at the queue Ex: Single queue Exponential interarrival time Exponential service time M/M/1 queueserver

13. M/M/1 Markov Chain State transition diagram if a birth-death process:

14. M/M/1 Flow Equations P i – probability of being in state i Flow from state n-1 to state n: P n-1 Flow from state n to state n-1:  P n P n-1 =  P n P n =  P n-1 =  n P 0

15. M/M/1 P 0 = 1/  n  < 1, 1/  n converges to 1-  P n =  n (  )

16. M/M/1 Average number of jobs in the system: E[n] = nP n =  /(1-  Response time:r = = Average number of jobs waiting in the queue: q =  2 /(1-  ) Average waiting time in the queue: w =  /  (1-  ) Probability of finding k or more jobs in the system: Pr(K≥k) =  k

17. M/M/1 Ex: In a web server, the time between requests follow exponential distribution with mean time between requests of 8 ms. Time required by the server to process each request is also exponential distribution with average service time 5 ms. a. What is the average response time observed? b. How much faster must the server process requests to halve this average response time? c. How many queue entries are necessary to buffer requests so that no more than one request per billion is lost?

18. M/M/c c – number of servers – job arrival rate n – jobs in the system (both on server & in queue)  – service rate for each server queueserver...

19. M/M/c If all servers are busy, mean service rate is: c  If some servers are idle, mean service rate is:n  (n<c 

20. M/M/c Probability of finding n jobs in the system: (c  ) n P 0 / n! n=1,2,… c P n = (c  ) n P 0 / c! c n-c n>c Total traffic intensity:  =  c  (utilization of each server) Probability of all servers being busy:  = Pr(K≥c) = (c  ) c P 0 / c!(1-  )

21. M/M/c Average number of jobs in the system: E[n] = c  +  /(1-  ) Average number of jobs waiting in the queue: q = (  ) / (1-  ) Little’s law: r = 1/  c  Average time jobs wait in the queue: w =  c 

22. M/M/c Ex: How does the response time in M/M/1 queue example change if the number of servers in the system is 4?