1. Queuing Theory Important definition in Queuing Theory: We define various terms, which are used in our queue model. 1.Queue length: The number of customers waiting in the line at any time is defined as queue length. 2.Average length of queue: Average length of queue is defined by the number of customers in the queue per unit time 3.Waiting time : Waiting time is the time up to which a unit has to wait in the queue before it is taken into service.

2. 4. Service time: The time taken by the server for servicing of a unit is called service time. 5. Busy Period: The time during which a server remains busy in servicing is called busy period. Therefore, the busy period is the time between the start of service of the first unit to the end of service of the last unit in the queue. 6. Idle period: The idle period of a server is the time during which it remains free because there is no customer present in the system. Therefore, the idle period of a server begins when all the customers in the queue are served and it continues upto the time of arrival of the customer.

3. 7. Mean arrival rate: the mean arrival rate in a waiting line situation is defined as the expected number of arrivals occurring in a time interval of length unity. 8. Mean service rate : The mean service rate is defined as the expected number of service completed in a time interval of length unity.

4. Queuing System A queuing system can be completed described by mainly five components a)Input source (arrival pattern) b)Queuing process c)Queue discipline d)Service mechanism (Service pattern) e)Customer’s behavior

5. Queue discipline The queue discipline is the rule, in which the manner of the customer’s behavior while waiting and the manner in which they are chosen for service. Some discipline are  First come, first served (FCFS) : If the customers are served in the order of their arrival, then it is known as first come, first served. For example, at cinema ticket window, railway reservation centre etc.

6.  Last come, first served (LCFS) : last arrival in the system is referred to servicing of the customer in an order just reserve of their arrival, so that the one who joins the queue in the last is serviced first. For example, in a big godown, where the last item loaded is removed first.  Selection in random order(SIRO) : In this rule, the customers are selected for service at random irrespective of their arrivals in the service system.

7.  Priority service : under this rule, the service is of two types a)Pre-emptive b)Non-pre-emptive. In Pre-emptive, the high priority customers are given service over the low priority customers and in Non-pre-emptive, the low priority customers are serviced before the high priority customers

8. Queuing Theory Queue are formed when waiting arises. Resources are less demand is more. Characteristics of queuing : 1.Arrival distribution: Inter arrival time is a period between two successive arrivals. Arrival rate (λ) : no. of customers arriving per unit time mean vale is denoted as µ. 2.Service distribution : service time is a time period between two successive services. Service rate (λ) : no. of customers served per unit time. Mean value is denoted as µ.

9. 3. Service channel : there can be one or more service channels. 4. Service Discipline (order of service) : a)FCFS b)LCFS c)SIRO (Service in random order) d)Priority. 5. Queue size : Queue may have a finite size as in the buffer area.

10. M/M/1 (Markovian ) Queue Now we turn to the analysis of waiting time. We wish to investigate the distribution of waiting times for message and the total time a message spends in the system, both waiting and being transmitted. This system is called an M/M/1 queue. The M stands for Markovian, after the mathematician Markov. The first M indicated that the arrival are independent of one another; that is the inter arrival times are distributed exponentially.

11. The second M indicates that the service times are also exponentially distributed. The 1 indicates that there is 1 server. The system can be analyzed in term of its state, specifically the state of the server(either idle or busy)and the state of the queue(number of message waiting).each time a message arrives or departs the system changes state.

12. Let, – n = number of message in the system(in queue plus in service) – A n = Arrival rate of message given ‘n’ in the system(message/sec) – D n →Departure rate of messages(messages/sec) – P n → Steady state probability of n messages in the system – G → Inter arrival rate The generalized model drives P n as a function of A n and D n. These probabilities are then used to determine the performance of system.

13. SERVICE TIME (transmission time) The departure rate of message (D n ) is related to the service time(T S ),which is in turn related to the message length(L) and channel speed B bits/sec. [ T s ] service time = T s = bit/sec. Average rate of departure = D n = Service Utilization It is defined as fraction of time the server is busy. As arrival rate increases,SU increases, departure rate increases;SU decreases. u = A/D

14. Important Formulas a) Average service time: T S = b) Arrival rate : Arrival rate(A) is denoted as msg/sec c) Service rate: Service rate = d) Utilization of server: u = A/D utilization is always a dimensionless quantity.

15. (e) Probability that there are 2 message in the system is given by, P 2 =(1-u)u 2 (f) the average number of message in the queue: Q = u 2 / (1-u) (g) the average number of message in the system is given by, N = u/(1-u) (h) the average waiting time(time in queue): T w = (i)The average time in the system “T”: T = T S +T w

16. Ex.9.16.1: Messages independently arrive to a system at the rate of 10 per minutes. Their lengths are exponentially distributed with an average of 3600 characters. They are transmitted on a 9600bps channel. A character is 8 bit long. Soln: a) Average service time: T S = = = 3sec b) Arrival rate : Arrival rate(A) = 10 msg/min = 10/60 = 1/6 A = 1/6 msg/sec c) Service rate: Service rate = D = 1/3 msg/sec d) Utilization of server: u = A/D = = = 0.5 utilization is always a dimensionless quantity.

17. (e) Probability that there are 2 message in the syatem is given by, P 2 =(1-u)u 2 = (1-0.5)(0.5) 2 = 0.125 (f) the average number of message in the queue: Q = u 2 / (1-u) = 0.5 (g) the average number of message in the system is given by, N = u/1-u = 0.5/1 – 0.5 N = 1 Note : N = Q + u (h) the average waiting time(time in queue): T w = = = 3 sec. (i)The average time in the system “T”: T = T S +T w = 3+3 sec = 6.0 sec.

18. Ex 9.16.2: Messages independently arrive to a system at a rate of 10 per min.Their lengths are exponentially distributed with an average of 4800 characters. They are transmitted on a 9600bps channel. A character is 8 bit long. Soln: Average message time: T S = = = 4 secT S. Arrival rate: A = 10 msg/min = 10/60 = 1/6 A = 1/6 msg/sec Service rate: D = = msg/sec

19. Utilization of server: u = = = = 2/3 The probability that there are 3 message in the system is given by, P 3 = (1-u)u 3 = (1- ) × (2/3) 3 = × = 8/71 The average number of message in the system is given by: N = = = N = 2

20. The average waiting time (time in queue): T w = = = × = 8 sec The average time in the system ‘T’: T = T w + T s = 8+4 = 12 sec.

21. Ex 9.16.3: in a M/M/1 queueing system,the average inter arrival ti9me between messages is 10sec, and the service time is 5 sec calculate. (a)Average rate of arrival (b) departure rate (c) channel utilization Soln: given G = 10sec, T S = 5sec A = = 0.1 msg/sec D = = = 0.2 msg/sec u = = = 0.5

22. T w = = ×5 = 5 sec. P 2 = (1-u)u 2 = 0.5 (o.5) 2 = 0.125 Q = = = 0.5 N = = 1 T = T s + T w = 10 sec

23. Q.1 Messages independently arrive to a system at the rate of 10 per minutes. Their lengths are exponentially distributed with an average of 2400 characters. They are transmitted on a 1200bps channel. A character is 8 bit long. Calculate Average service time, Arrival rate, Service rate, Utilization of server, Probability of the message in the system, The average number of message in the queue, the average number of message in the system, the average waiting time time in queue), The average time in the system “T”. Q.2 Messages independently arrive to a system at the rate of 10 per minutes. Their lengths are exponentially distributed with an average of 6000 characters. They are transmitted on a 9600bps channel. A character is 8 bit long. Calculate Average service time, Arrival rate, Service rate, Utilization of server, Probability of the message in the system, The average number of message in the queue, the average number of message in the system, the average waiting time time in queue), The average time in the system “T”.

24. Ques 3 Messages independently arrive to a system at the rate of 10 per minutes. Their lengths are exponentially distributed with an average of 5800 characters. They are transmitted on a 9600bps channel. A character is 8 bit long. Calculate Average service time, Arrival rate, Service rate, Utilization of server, Probability of the message in the system, The average number of message in the queue, the average number of message in the system, the average waiting time time in queue), The average time in the system “T”.