4.4 The Kolmogoroff Differential Equations.
Like the Chapman-Kolmogoroff equations, the Kolmogoroff differential equations are simplest to write
down in matrix form. They say that the transition matri
4.2 Markov Processes.
Let's summarize the assumptions from section 4.1 that we will be making in the rest of this chapter. We
have a system that at any non-negative time t can be in any of N states nu
5.2 Multiple Server Queues.
The previous section considered single server queues. In this section we extend the analysis to queues
with c servers. This is often called an M/M/c queue. We let
= arriva
4
Continuous Time Markov Processes
4.1 Basic Principles.
Many systems in the real world are more naturally modeled in continuous time as opposed to discrete
time. For example, if we are modeling the n
3
Continuous Probability
3.1 Probability Density Functions.
So far we have been considering examples where the outcomes form a finite or countably infinite set. In
many situations it is more natural t
3.2 Joint Probability Distributions.
Recall that in section 1.9 we discussed two ways to describe two or more discrete random variables.
We could either use the joint probability mass function or cond
4.7 Revenues and Costs.
In the previous sections we saw how to compute the transition matrices P(t) = etR where R is the
generator matrix and we saw that P(t) approaches a limiting matrix P() as t ten
4.6 Steady State Probablilites.
In the previous section we saw how to compute the transition matrices P(t) = etR where R is the
generator matrix. We saw that each element of P(t) was a constant plus a
3.3 Sums and Other Functions of Random Variables.
We often form new random variables by performing the usual mathematical operations to existing
random variables. In this case we may be interested in
4.5 Solving Kolmogoroff's Equation The Matrix Exponential.
In the last section we discussed the Kolmogoroff differential equations which say that the transition
matrices P(t) satisfy
(1)
= P(t)R
(2)
=