2. M / M / 1 M - Memoryless* arrival process M - Memoryless* service time distribution 1 - One server * Memoryless / Deterministic / General Queueing System Nomenclature...

3. Queueing System Nomenclature Examples: M / M / 1 M / D / 2 G / D / 

4. Notation to Describe a Queueing System Service time distribution Interarrival time distribution Number of servers M=exponential distribution D=degenerate distribution E k =Erlang distribution(shape parameter=k) G=general distribution

5. Load diagram of a simple queue

6. Processes at work in a queue A(t) : no. of arrivals by time t L(t) : no. of departures by time t  (t) : load on the system by time t (no. of jobs in system x time spent by each job)  (T) = area between A(T) and L(T) = d 1 + d 2 + …..+ d n N(t) = avg. no of jobs in system =  (t) /t

7. Performance measures N(t): avg. no. of jobs in system N q (t): avg. no. of jobs in queue R(t): avg. time spent in system (per job) W(t): avg. time spent in queue (per job) These measures are inter-related These measures will depend on type of queue (factors like: arrival pattern, service times, queue discipline, no. of servers, etc) However, a basic relationship holds for almost all queues

8. Little’s Relation Avg. no. N(t) =  (t) /t Arrival rate (t) = A(t)/t Avg. time R(t) =  (t) /A(t) (for large n)  N(t) = [  (t) /A(t) ][A(t)/t] = R(t) (t) Now let t   For arrival processes with constant rate (t)  as t   (e.g. Poisson process, why ?) and R(t)  R (to be proved) so : lim t   N(t) = N = R Little’s Relation: N = R Holds for any queue discipline, can be applied to any sub-system

9. Application of Little’s Relation to sub-systems Look at the ‘queue’, or buffer as a sub-system Arrival rate into ‘queue’ = Avg. no. in the ‘sub-system’ = N q Avg. time spent in ‘sub-system’ = W By Little: N q = W

10. Application of Little’s Relation to sub-systems P 0 = P( 0 jobs in queue) = P(server is free) Consider the server as a sub-queue. Can hold either 0 or 1 jobs (prob. P 0 or 1-P 0 ) expected no. of jobs N s = 0. P 0 + 1.(1-P 0 ) = 1- P 0 arrival rate into server = (in equilibrium state) avg. time spent in server = 1/  (  service rate) Little’ Relation: N s =. 1/  ( /  = , utilization)  P 0 = 1-  (for G/G/1 queue)

11. Another Application of Little Three classes of messages arrive at a service facility. Loads are such that no queueing is necessary. Class 1 messages arrive at 120 per minute, processing 200msec (avg) Class 2 messages arrive at 20 per minute, depart six per min (avg) Class 3 messages arrive at 10 per minute, depart after 30 sec (avg) What kind of system is it ? What is the average no. of messages in the system ?

12. Practicals and tutorials Tutorials: Mon 4-5 FSB322 (from today) Practicals: Tue 1-2 and 4-5, AL11 (starting next week) 1-2 hour: 4-5 hour: Practicals: 10% of final grade Homeworks: 10% of final grade

13. Booklet available It contains: –Past exam papers –References –Tables –Introduction to Taylor and tutorials Booklet is required for practicals