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Queuing Models

Queuing Models. Dr. Mahmoud Alrefaei. Introduction. Each one of us has spent a great deal of time waiting in lines. One example in the Cafeteria. Other examples of queues are Printer queue Customers in front of a cashier Calls waiting for answer by a technical support.

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Queuing Models

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  1. Queuing Models Dr. Mahmoud Alrefaei

  2. Introduction • Each one of us has spent a great deal of time waiting in lines. • One example in the Cafeteria. • Other examples of queues are • Printer queue • Customers in front of a cashier • Calls waiting for answer by a technical support

  3. What makes up a queue? • The System: A collection of objects under study. • It is important to define the system boundaries. • The Entities: The people, organisms, or objects that enter the system requiring some kind of service. • The Servers: The people, organisms, or machines that perform the service required. • The Queue: An accumulation of entities that have entered the system but have not been served.

  4. Types of Queues Single Stage

  5. Multiple Stage - Manufacturing Plant

  6. Multi-channel Single Stage - Bank

  7. Parallel Single Stage - Supermarket

  8. Customer Discrimination – Bus station

  9. Converging Arrivals - Walk-in and Drive-thru

  10. Queue Discipline • First Come First Served - FCFS • Most customer queues. • Last Come First Served - LCFS • Packages, Elevator. • Served in Random Order - SIRO • Entering Buses • Priority Service • Multi-processing on a computer. • Emergency room.

  11. What factors affect system performance • The Arrivals Process. • The time between any two successive arrivals • Does this depend on the number of people in the system? • Finite populations. • The Service Process. • The time taken to perform the service. • Does this depend on the number of people in the system? • The number of servers operating in system.

  12. Measuring System Performance • The total time an “entity” spends in the system (Denoted by W) • The time an “entity spends in the queue. (Denoted by Wq) • The number of “entities” in the system. (Denoted by L) • The number of “entities” in the queue. (Denoted by Lq)

  13. Measuring System Performance • The percentage of time the servers are busy (Utilization time) • These quantities are variable over time.

  14. Poisson Process • Let {N(t), t>0} be the number of customers arrive until the time t • {N(t), is said to be a Poisson Process having rate , for >0, if • N(0) = 0 • N(t+s) – N(t) Does not depend on the previous history • N(t+s) – N(t) is independent of t.

  15. Poisson Process The number of events in any interval of length t is Poisson distributed with mean t. That is for all s, t > 0 and n=0,1,2,...

  16. Inter-Arrival Times • What are inter-arrival times? • Ti is the time between the (i-1)-st and the i-th events. • Poisson Process can be used for modeling arrival process T3 T2 T1 S1 S2 S3

  17. Birth-Death Processes • A birth-death process is used to model populations of entities in a system • The state of the system at time t is the number of entities in the system at that time, often denoted by N(t). • Births and deaths occur at a constant rate (like the Poisson process model)

  18. Birth-Death Processes Pi,j(t) is defined to be the probability that there are j entities in the system at time t, given that there were i entities in the system at time 0. 0 1 2 j-1 j j+1 ... ...

  19. Birth-Death Processes • The state of the system must be a non-negative integer • Law 1 • A birth increases the state from j to j+1 • The variable j is called the birth rate for state j • A birth occurs between times t and t + t with probability jt + o(t)

  20. Birth-Death Processes • Law 2 • A death decreases the state from j to j-1 • The variable j is called the death rate for state j (note that 0=0) • A death occurs between times t and t + t with probability jt + o(t ) • Law 3 • Births and Deaths are independent

  21. Birth-Death Processes • Can more than one event happen between t and t + t? • Why must 0=0? • Knowledge of j and j completely specifies a Birth-Death process.

  22. Birth-Death Processes • Birth-Death Processes can be used to model most M/M/... queuing systems. • An arrival is considered a “birth”. • A service completion is considered a “death”. • Let Pi,j(t) be the probability N(t+s)–N(s)=j given that N(s)=i (or N(t)=j given N(0)=i • It turns out that for many queuing systems, Pi,j(t) will approach a limit j as t gets larger.

  23. Birth-Death Processes • This limit will be independent of the initial state i. • j is called the steady state or equilibrium probability of state j. • j can be thought of as the probability that at some instant in the future there are j entities in the system. • j can also be thought of as the fraction of time that there are j entities in the system.

  24. Exponential Distribution • The exponential distribution is characterized by

  25. Exponential Distribution • What is P(T>t) for an exponential distribution with parameter ? e- t • P(T>t+s|T>s) is the probability of waiting a further time t after having already waited to time s. • What is P(T>t+s|T>s) for an exponential distribution with parameter ? e- (t+s) / e- s = e- t • Answer: P(T>t+s|T>s) = P(T>t) This is called the memoryless property

  26. An Example of Memoryless • Suppose that the amount of time one spends in a bank is exponentially distributed with mean ten minutes. • What is ? • What is the probability that a customer will spend more a quarter of an hour in the bank? • You have been waiting for ten minutes already. Now what is the probability that you will spend more than a quarter of an hour in the bank? • What has a lack of memory, you or the distribution?

  27. Birth-Death Processes • Consider a M/M/1 queuing system. • Inter-arrival times are exponential with rate . • Service times are exponential with rate . • Suppose there are j entities in the system at time t. What is the probability of an arrival in the interval (t,t + t]? Hint: use Taylor series expansion on F(t + t)-F(t) = 1 - e- t =  t + o(t)

  28. Birth-Death Processes • So the arrivals and service completions of a M/M/1 queue are a birth-death process with the following rate diagram 0 1 2 j-1 j j+1 ... ...

  29. Birth-Death Processes (Balance equations) • It can be shown by induction that • These are called balance equations • Therefore:

  30. Birth-Death Processes • If we define the constants • then • and we know that

  31. Birth-Death Processes • So • This means that the infinite sum of the cj’s must converge. • If this sum is infinite then no steady-state distribution can exist.

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