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Queuing Theory

Queuing Theory. Queuing Theory. Queuing Theory deals with systems of the following type: Typically we are interested in how much queuing occurs or in the delays at the servers. Server Process( es ). Input Process. Output. Queuing Theory Notation.

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Queuing Theory

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  1. Queuing Theory

  2. Queuing Theory • Queuing Theory deals with systems of the following type: • Typically we are interested in how much queuing occurs or in the delays at the servers. Server Process(es) Input Process Output

  3. Queuing Theory Notation • A standard notation is used in queuing theory to denote the type of system we are dealing with. • Typical examples are: • M/M/1 Poisson Input/Poisson Server/1 Server • M/G/1 Poisson Input/General Server/1 Server • D/G/n Deterministic Input/General Server/n Servers • E/G/Erlangian Input/General Server/Inf. Servers • The first letter indicates the input process, the second letter is the server process and the number is the number of servers. • (M = Memoryless = Poisson)

  4. Definition of a queueing system Departure of servedcustomers Customer arrivals Departure of impatient customers • A queueing system can be described as follows: • "customers arrive for a given service, wait if the service cannot start immediately and leave after being served" • The term "customer" can be men, products, machines, ...

  5. History of queueing theory • The theory of queueing systems was developed to provide models for forecasting behaviors of systems subject to random demand. • The first problems addressed concerned congestion of telephone traffic (Erlang, "the theory of probabilities and telephone conversations ", 1909) • Erlang observed that a telephone system can be modeled by Poisson customer arrivals and exponentially distributed service times • Molina, Pollaczek, Kolmogorov, Khintchine, Palm, Crommelin followed the track

  6. Interests of queueing systems • Queuing theoryfoundnumerousapplications in: • – Trafic control (communication networks, air traffic, …) • – Planing (manufacturingsystems, computer programmes, …) • – Facilitydimensioning (factories, ...)

  7. Classification of queueing systems

  8. Characteristics of simple queueing systems • Queueingsystemscanbecharacterizedwithseveralcriteria: • Customer arrivalprocesses • Service time • Service discipline • Service capacity • Number of service stages

  9. Notation of Kendall The following is a standard notation system of queueing systems T/X/C/K/P/Z with – T: probability distribution of inter-arrival times – X: probability distribution of service times – C: Number of servers – K: Queue capacity – P: Size of the population – Z: service discipline

  10. Customer arrival process T/X/C/K/P/Z • T can take the following values: • – M : markovian (i.e. exponential) • – G : general distribution • – D : deterministic • – Ek : Erlang distribution • – … • • If the arrivals are grouped in lots, we use the notation T[X] where X is the random variable indicating the number of customers at each arrival epoch • – P{X=k} = P{k customers arrive at the same time} • • Some arriving customers can leave if the queue is too long

  11. Service times T/X/C/K/P/Z • X cantake the following values: • – M : markovian (i.e. exponential) • – G : general distribution • – D : deterministic • – Ek : Erlang distribution • – … k exponential servers with parameter m Erlang distribution Ekwithparameterm

  12. Number of servers T/X/C/K/P/Z In simple Queuing Systems, servers are identical

  13. Queue capacity T/X/C/K/P/Z Loss of customers if the queue is full Capacity K

  14. Size of the population T/X/C/K/P/Z The size of the population canbeeitherfinite or infinite For a finite population, the customerarrival rate is a function of the number of customers in the system: l(n).

  15. Service discipline T/X/C/K/P/Z • Z can take the following values: • FCFS or FIFO : First Come First Served • LCFS or LIFO : Last Come First Served • RANDOM : service in random order • HL (Hold On Line) : when an important customer arrives, it takes the head of the queue • PR ( Preemption) : when an important customer arrives, it is served immediately and the customer under service returns to the queue • PS (Processor Sharing) : All customers are served simultaneously with service rate inversely proportional to the number of customers • GD (General Discipline)

  16. Previous arrival Begin Service End Service Arrival Time  w s r •  = interarrival time •  = mean arrival rate = 1/E[] • s = service time per job •  = mean service rate per server = 1/E[s] (total service rate for m servers is m) • nq = number of jobs waiting to receive • service. • ns = number of jobs receiving service • n = number of jobs in system • n = nq + ns • r = response time = w + s • w = waiting time Note that all of these variables are random variables except for  and  . Chapter 30 Introduction to Queueing Theory

  17. Rules for All Queues • Stability Condition • If the number of jobs becomes infinite, the system is unstable. For stability, the mean arrival rate less than the mean service rate.  < m • Does not apply to finite buffer system or the finite population systems They are always stable. • Finite population: queue length is always finite. • Finite buffer system: arrivals are lost when the number of jobs in the system exceed the system capacity. Chapter 30 Introduction to Queueing Theory

  18. Rules for All Queues • Number in System vs. Number in Queue • n = nq + ns • E[n] = E[nq]+E[ns] • Also, if the service rate of each server is independent of the number in queue • Cov(nq,ns) = 0 • Var[n] = Var[nq]+Var[ns] Chapter 30 Introduction to Queueing Theory

  19. Rules for All Queues • Number vs. Time (Little’s law) • If jobs are not lost due to buffer overflow, the mean jobs is related to its mean response time as follows: mean number of jobs in system = arrival rate x mean response time • Similarly mean jobs in queue = arrival rate x mean waiting time • For finite buffers can use effective arrival rate, that is, the rate of jobs actually admitted to the system. Chapter 30 Introduction to Queueing Theory

  20. Rules for All Queues • Time in System vs. Time in Queue • Time spent in system, response time, is the sum of waiting time and service time r = w + s • In particular: E[r] = E[w] + E[s] • If the service rate is independent of jobs in queue • Cov(w,s) = 0 • Var[r] = Var[w] + Var[s] Chapter 30 Introduction to Queueing Theory

  21. Proof of Little’s LawMean jobs in system = arrival rate x mean response time Chapter 30 Introduction to Queueing Theory

  22. Proof of Little’s Law J: total time spent inside the system by all three jobs (the hatched area) 30.4 (a): Mean time spent in the system = J/N 30.4 (b): Mean number in the system = J/T = J/N * N/T = Mean time spent in the system * Mean arrival rate Chapter 30 Introduction to Queueing Theory

  23. Applying Little’s Law • Example: • A disk server satisfies an I/O request in average of 100 msec. I/O rate is about 100 requests/sec. What is the mean number of requests at the disk server? • Mean number at server = arrival rate x response time = (100 requests/sec) x (0.1 sec) = 10 requests Chapter 30 Introduction to Queueing Theory

  24. Types of Stochastic Processes • A stochastic process is a family of random variables indexed by the parameter t, such as time. e.g.) Number of jobs at CPU of computer system at time t is a random variable n(t) • Time and state can be discrete or continuous Chapter 30 Introduction to Queueing Theory

  25. Types of Stochastic Processes • Markov Process • If future states depend only on the present and are independent of the past (memoryless) then called markov process • A discrete state Markov Process is a Markov chain • M/M/m queues can be modeled using Markov process • time spent by a job – Markov process • the number of jobs - Markov chain Chapter 30 Introduction to Queueing Theory

  26. Types of Stochastic Processes • Birth-Death Process • The Markov chain in which transitions restricted to neighboring states. • Can represent states by integers, s.t. process in state n can only go to state n+1 or n-1 • Arrival (birth) causes state to change by +1 and departure after service (death) causes state to change by –1 • Only if arrive individually, not in bulk Chapter 30 Introduction to Queueing Theory

  27. Types of Stochastic Processes • Poisson Processes • If inter arrival times are IID and exponentially distributed, then number of arrivals over interval [t,t+x] has a Poisson distribution  Poisson Process • Popular in queuing theory because arrivals are then memory less Chapter 30 Introduction to Queueing Theory

  28. Properties of Poisson Process • Merging k Poisson streams with mean rate i gives another Poisson stream with mean rate  = i (b) If Poisson stream split into k substreams with probability pi, each substream is Poisson with mean rate pi Chapter 30 Introduction to Queueing Theory

  29. Properties of Poisson Process (c) If arrivals to single server with exponential service times are Poisson with mean , departures are also Poisson with mean , if (<) (d) Same relationship holds for m servers as long as total arrival rate less than total service rate Chapter 30 Introduction to Queueing Theory

  30. Markov Processes Birth-death Processes Poisson Processes Types of Stochastic Processes Chapter 30 Introduction to Queueing Theory

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