Queueing Theory: Part I

1. Queueing Theory: Part I

2. The Basic Structure of Queueing Models The Basic Queueing Process: Customers are generated over time by an input source. The customers enter a queueing system. A required service is performed in the service mechanism.

3. Input source Queueing system Served customers Service mechanism Queue Customers

4. Input Source (Calling Population): The size of Input Source (Calling Population) is assumed infinite because the calculations are far easier. The pattern by which customers are generated is assumed to be a Poisson process.

5. The probability distribution of the time between consecutive arrivals is an exponential distribution. The time between consecutive arrivals is referred to as the interarrival time.

6. Queue: The queue is where customers wait before being served. A queue is characterized by the maximum permissible number of customers that it can contain. The assumption of an infinite queue is the standard one for most queueing models.

7. Queue Discipline: The queue discipline refers to the order in which members of the queue are selected for service. For example, (a) First-come-first-served (b) Random

8. Service Mechanism: The service mechanism consists of one or more service facilities, each of which contains one or more parallel service channels, called servers. The time at a service facility is referred to as the service time. The service-time is assumed to be the exponential distribution.

9. Elementary Queueing Process Served customers Queueing system Queue C C C C S S Service S facility S Customers C C C C C C C Served customers

10. Distribution of service times Number of servers Distribution of interarrival times Where M = exponential distribution (Markovian) D = degenerate distribution (constant times) = Erlang distribution (shape parameter = k) G = general distribution(any arbitrary distribution allowed)

11. Both interarrival and service times have an exponential distribution. The number of servers is k . Interarrival time is an exponential distribution. No restriction on service time. The number of servers is exactly 1.

12. Terminology and Notation State of system = # of customers in queuing system. Queue length = # of customers waiting for service to begin. N(t) = # of customers in queueing system at time t (t 0) = probability of exactly n customers in queueing system at time t.

13. # of servers in queueing system. A mean arrival rate (the expected number of arrivals per unit time) of new customers when n customers are in system. A mean service rate for overall system (the expected number of customers completing service per unit time) when n customers are in system. Note: represents a combined rate at which all busy servers (those serving customers) achieve service completions.

14. When is a constant for all n, it is expressed by When the mean service rate per busy server is a constant for all n 1, this constant is denoted by . Under these circumstances, and are the expected interarrival time and the expected service time. is the utilization factor for the service facility.

15. The state of the system will be greatly affected by the initial state and by the time that has since elapsed. The system is said to be in a transient condition. After sufficient time has elapsed, the state of the system becomes essentially independent of the initial state and the elapsed time. The system has reached a steady-state condition, where the probability distribution of the state of the system remains the same over time.

16. The probability of exactly n customers in queueing system. The expected number of customers in queueing system The expected queue length (excludes customers being served)

17. A waiting time in system (includes service time) for each individual customer. A waiting time in queue (excludes service time) for each individual customer.

18. Relationships between and Assume that is a constant for all n. In a steady-state queueing process, Assume that the mean service time is a constant, for all It follows that,

19. The Role of the Exponential Distribution

20. An exponential distribution has the following probability density function:

21. Relationship to the Poisson distribution Suppose that the time between consecutive arrivals has an exponential distribution with parameter . Let X(t) be the number of occurrences by time t (t 0) The number of arrivals follows (11.1) & (11.2)

22. The Birth-and-Death Process Most elementary queueing models assume that the inputs and outputs of the queueing system occur according to the birth-and-death process. In the context of queueing theory, the term birth refers to the arrival of a new customer into the queueing system, and death refers to the departure of a served customer.

23. The birth-and-death process is a special type of continuous time Markov chain. State: 0 1 2 3 n-2 n-1 n n+1 and are mean rates.

24. Rate In = Rate Out Principle. For any state of the system n (n = 0,1,2,…), average entering rate = average leaving rate. The equation expressing this principle is called the balance equation for state n.

25. Rate In = Rate Out State 0 1 2 n – 1 n

26. State: 0: 1: 2: To simplify notation, let for n = 1,2,…

27. and then define for n = 0. Thus, the steady-state probabilities are for n = 0,1,2,… The requirement that so that implies that

28. The definitions of L and specify that is the average arrival rate. is the mean arrival rate while the system is in state n. is the proportion of time for state n,

29. The M/M/s Model A M/M/s model assumes that all interarrival times are independently and identically distributed according to an exponential distribution, that all service times are independent and identically distributed according to another exponential distribution, and that the number of service is s (any positive integer).

30. In this model the queueing system’s mean arrival rate and mean service rate per busy server are constant ( and ) regardless of the state of the system. (a) Single-server case (s=1) State: 0 1 2 3 n-2 n-1 n n+1

31. (b) Multiple-server case (k > 1) for n = 0,1,2,… for n = 1,2,…k for n = k, k+1,... State: 0 1 2 3 k-2 k-1 k k+1

32. When the maximum mean service rate exceeds the mean arrival rate, that is, when a queueing system fitting this model will eventually reach a steady-state condition.

33. When , the mean arrival rate exceeds the mean service rate, the preceding solution “blows up” and grow without bound. Assuming , we can derive the probability distribution of the waiting time in the system (w) for a random arrival when the queue discipline is first-come-first-served. If this arrival finds n customers in the system, then the arrival will have to wait through n + 1 exponential service time, including his or her own.

34. [1] Single-Server case ( k = 1) Birth-Death Process 0 1 2 n-1 n n+1 Rate In = Rate Out State 0 1 2 n

39. Similarly,

42. Example

43. [2] Multiple-Server case ( k > 1) Birth-Death Process 0 1 2 3 k-2 k-1 k k+1

44. Rate In = Rate Out State 0 1 2 k - 1 k k + 1