1. Queuing Theory Summary of results

2. 2 Notations Typical performance characteristics of queuing models are: L : Ave. number of customers in the system L Q : Ave. number of customers waiting in queue W : Ave. time customer spends in the system W Q : Ave. time customer spends waiting in the queue

3. 3 Queue notation M/M/k/c Arrival process M = Markovian GI = General Departure process (Service time distribution) M = Markovian G = General Number of servers Capacity of the queue If nothing is specified, we assume infinite capacity

4. 4 M/M/1 queue

5. 5 M/M/1/N queue

6. 6 Expected amount of time spent by the customer…. But, what is a customer? Do we include those who came but didn’t join the queue because it was full? Or, are we including only those who actually joined the system? Depending on our consideration, λ a = λ in the first case; whereas λ a =λ(1-P N ) in the second case. Then,

7. 7 Tandem queues M/M/1

8. 8 M/G/1 Derivation of these results is slightly more tedious because, unlike previous models where the Markov theory was extensively used, M/G/1 model requires Renewal Reward theory.

9. 9 M/G/1: Priority queues Let there be two types of customers (Type 1 and 2). Type 1 being the priority class. Meaning, service can never begin for Type 2 customer, if Type 1 is waiting in queue. However, preemption is not allowed. Let arrival rate of Type 1 customers be λ 1 and that of Type 2 be λ 2 (both arrivals are Poisson processes). Respective service time distribution be G 1 and G 2. That is S 1 ~ G 1 and S 2 ~ G 2. Once again, we derive the performance characteristics based on the renewal reward theory.

10. 10 M/G/1: Priority queues

11. 11 M/M/k queues Two types of queues could be considered: 1.Loss function queues: Customer does not join the system if (s/)he sees all k servers busy. This queue is called Erlang’s Loss system. 2.Infinite capacity queues. These are exact extensions of M/M/1 queues.

12. 12 M/M/k queues