Introduction l Definition l MM queues l MM1

Introduction l Definition l M/M queues l M/M/1 l M/M/S l M/M/infinity l M/M/S/K 1

Queuing system l A queuing system l l is a place where customers arrive l According to an “arrival process” l To receive service from a service facility Can be broken down into three major components l The input process l The system structure l The output process Customer Population Waiting queue Service facility 2

Characteristics of the system structure λ l μ Queue l l Infinite or finite Service mechanism l l λ: arrival rate μ: service rate 1 server or S servers Queuing discipline l FIFO, LIFO, priority-aware, or random 3

Queuing systems: examples l Multi queue/multi servers l l Example: l Supermarket l Blade centers § orchestrator . . . Multi-server/single queue l Bank l immigration 4

Kendall notation l David Kendall l A British statistician, developed a shorthand notation l To describe a queuing system l A/B/X/Y/Z § A: Customer arriving pattern § B: Service pattern § X: Number of parallel servers § Y: System capacity § Z: Queuing discipline M: Markovian D: constant G: general Cx: coxian 5

Kendall notation: example l M/M/1/infinity l l A queuing system having one server where l Customers arrive according to a Poisson process l Exponentially distributed service times M/M/S/K K l M/M/S/K=0 l Erlang loss queue 6

Special queuing systems l Infinite server queue λ l . . μ Machine interference (finite population) S repairmen N machines 7

M/M/1 queue λ l μ λ: arrival rate μ: service rate λn = λ, (n >=0); μn = μ (n>=1) 8

Traffic intensity l rho = λ/μ l l It is a measure of the total arrival traffic to the system l Also known as offered load l Example: λ = 3/hour; 1/μ=15 min = 0. 25 h Represents the fraction of time a server is busy l In which case it is called the utilization factor l Example: rho = 0. 75 = % busy 9

Queuing systems: stability l λ<μ l l N(t) busy => stable system l 3 2 1 1 λ>μ idle 2 3 4 5 6 7 8 9 10 11 Time Steady build up of customers => unstable N(t) 3 2 1 1 2 3 4 5 6 7 8 9 10 11 Time 10

Example#1 l A communication channel operating at 9600 bps l l Receives two type of packet streams from a gateway l Type A packets have a fixed length format of 48 bits l Type B packets have an exponentially distribution length § With a mean of 480 bits If on the average there are l l 20% type A packets and 80% type B packets Calculate the utilization of this channel l Assuming the combined arrival rate is 15 packets/s 11

Performance measures l L l l Lq l l Mean queue length in the queue space W l l Mean # customers in the whole system Mean waiting time in the system Wq l Mean waiting time in the queue 12

Mean queue length (M/M/1) 13

Mean queue length (M/M/1) (cont’d) 14