ts reserved. Any use is subject to the Terms of Use as given at the website. Traffic Engineering for Voice and Data Traffic Engineering for Voice and Data New call behavior is completely independent of the state the telephone system, SO the rate of changing from state j to j + 1 is a constant value, 1, for every state j. The rate of changing from state j to j - 1, on the other hand, depends on the number of calls in the system. Each call in the system has a rate of ending. Since state j has j calls in progress, the total rate of a call ending in state j is j times , and therefore, the rate of the system going from state j to j - 1 is ju. The state diagram is shown in Fig. 23.1. The arrows indicate the rates between states. If we think of each state as having some probability Pj, then we can picture the probabilities as functions of time Pi(t) flowing along the arrows in the state diagram and generating a set of time derivatives in Eq. (23.3): P(t) = (23.3) (j+1)uPj+1(t + XPj-1(t)-juPj(t)-XP(t)for Let us look at these differential equations for a moment to understand what they are telling us. Suppose that we have some knowledge of the system's state at a specific time t = 0. For example, we may know with 100 percent certainty that there are no calls on the system when we first turn it on, so Po(0) = 1.0 and P(0) = 0.0, for j > 0. Or our knowledge of the system may be statistical in nature, that each state j has some prob- ability P(0) of being occupied by the system at the specific time t = 0. Once we have a set of probabilities for each state, we can computationally integrate the differential equations to get these probabilities over a period of time. This is the same sort of cal- culation early astronomers used to integrate the differential equations of gravity and inertia to predict planetary motion. The equilibrium case occurs when the probabilities are not changing as time passes, when all their time derivatives are zero. When P(t) = 0, Eq. (23.3) reduces to the sim- pler Eq. (23.4). We can think of Eq. (23.4