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 The system structure The output process The service facility may contain more than one server, and it is assumed that a server can serve one customer at a time. If an arriving customer finds the service facility occupied, he joins the waiting queue. This customer will receive his service later in time, either when he reaches the head of the waiting queue or according to some service disciplines, and then leave the system upon completion of his service. The schematic diagram of a queuing system is depicted in the above presented figure. A queuing system is referred to as just a queue, or queuing node. Customer Population Waiting queue Service facility

Characteristics of the system structure λ: arrival rate μ: service rate λ μ Queue Infinite or finite Service mechanism 1 server or S servers Queuing discipline FIFO, LIFO, priority-aware, or random The service facility may comprise one or more servers. Queuing discipline refers to the way in which customers in the waiting queue are selected and served by the servers. In general we have: First Come First Served (FCFS), Last Come First Served (LCFS), priority, or random.

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

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 M: Markovian D: constant G: general Cx: coxian

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

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

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

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 To understand better the physical meaning of this unit, take a look at the traffic presented to a single resource. One Erlang is equivalent to a single user who uses that resources 100% of the time, or alternatively, 10 users who each occupy the resource 10% of the time. A traffic intensity greater than one indicates that customers arrive faster than they are served and is a good indication of the minimum number of servers required to achieve a stable system. For an example, a traffic intensity of 2.5 Erlangs indicates that at least three servers are required.

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

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

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

Mean queue length (M/M/1)

Mean queue length (M/M/1) (cont’d)