### Abstract

1. HW1 due today in class. No late homeworks, because it’s being graded tonight. 2. I need grading volunteers for tonight. Remember that you will also be designing one new interesting homework question. 3. You will get HW1 solutions on Wednesday. 4. HW2 will be on the web site today and is due in one week. 5. This week I will be teaching an EXTRA CLASS during the Friday recitation slot (1:30- 3:00 p.m.). This is to make up for the fact that I will be away Wednesday Sept. 30th. Little’s law is probably the single most famous queueing theory result. Little’s law states that the average number of jobs in the system is equal to the product of the average arrival rate into the system and the average time a job spends in the system. This also holds when the “system” consists of just the “queues ” in the system. Little’s law applies to both open and closed systems, and we’ll explain it in both cases. 2 Little’s Law for Open Systems Let’s first consider only open systems, as shown in Figure 1. Theorem 1 For any ergodic open system we have that: E {N} = λE {T} where E {N} is the expected number of jobs in the system, λ is the average arrival rate into the system, and E {T} is the mean time in system of jobs. It is important to notice that Little’s theorem makes no assumptions about the arrival process, the service time distributions at the servers, the network topology, the service order, nothing! At this point it may be hard to appreciate Little’s law. The usefulness of Little’s law stems from the fact that when we study Markov chains, we’ll see many techniques for computing E {N}. By applying Little’s law, this will immediately give us E {T}.

### Keyphrases

little law arrival rate average arrival rate hw1 solution friday recitation slot service time distribution little theorem average number closed system markov chain many technique mean time web site today extra class little law applies arrival process wednesday sept open system let average time homework question late homework open system network topology service order hw1 due today ergodic open system expected number little law state theory result