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Introduction

Introduction. Definition M/M queues M/M/1 M/M/S M/M/infinity M/M/S/K. Queuing system. A queuing system is a place where customers arrive According to an “arrival process” To receive service from a service facility Can be broken down into three major components The input process

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Introduction

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  1. Introduction • Definition • M/M queues • M/M/1 • M/M/S • M/M/infinity • M/M/S/K

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

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

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

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

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

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

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

  9. Traffic intensity • rho = λ/μ • It is a measure of the total arrival traffic to the system • Also known as offered load • Example: λ = 3/hour; 1/μ=15 min = 0.25 h • Represents the fraction of time a server is busy • In which case it is called the utilization factor • Example: rho = 0.75 = % busy

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

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

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

  13. Mean queue length (M/M/1)

  14. Mean queue length (M/M/1) (cont’d)

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