1. Probability Fundamentals
Version 1.2
Date:
We will talking about the elementary notions in Probability, especially related to our discussion in DEDS.
The memoryless triangle
The probability flow balance equation
The selected reference for this part is still CLB.
Copyright by Yu-Chi Ho

2. The Memoryless (Markov) Triangle
Constant
Arrival rate
Poisson
Arrival in
[0,t)
Exponential
Inter-arrival
time
This is the single most important fact to commit to memory.
Exponential Inter-arrival time: means the inter-arrival time distribution of the customers is exponential.
Constant arrival rate: means the probability of a single arrival in time (t) is equal to *(t) +o(t), where  is the so-called constant arrival rate and the arrivals are independent.
Poisson arrival in [0,t): means the probability of n arrivals in the interval [0,t) obeys the Poisson distribution.
The triangle means that we can get the other two properties from any one of the three.
This relationship can simplify our analysis, since each of them is a typical model in special fields.
We give a close view in the lecture notes this week (LN#1). You can also proceed to the referred sections in that notes.
Memoryless: if you loss and get mad, the same probability of getting a mad next time.
Copyright by Yu-Chi Ho

3. Probability Flow Equationj r
The probability flow equation is an important tool in the Queueing System analysis. Let us clarify the notations in the diagram above first.
Circle. Each circle represents one state of the system. Here, we use the number of the customers in the system to characterize the states.
Arrow. The arrow shows the transition of the states, i.e., from i to j. This is the State-Transition diagram of state j.
pij. It is the probability of the transition from state i to state j. We assume pij to be stationary.
qij. It is the transition rate from state i to j. We assume qij to be stationary.
i(t). It is the probability for the state being at i at time t.
Π(t). It is the probability vector, where Π(t)=[0(t), 1(t), …].
The probability flow equation tells us that the input flow of one state equals its output flow. This is only a intuitive explanation. See also the reading materials.
Secondary, let us consider the two summations in line 1 above.
The left one means the definition of pji is legal, since the probability of there existing a transition from state j to another state (including state j itself) is always 1.
The right one means the pure flow rate is zero in balance state. For details, see CLB p414.
Using the two equations above, we can finally get the probability flow equation (in line 3). We explain the representation of each item in equation 2 above.
First, i(t) means the probability fluid. qij means the probability flow rate from i to j.
The left part of the equation 2 means the net probability flow rate into state j.
The first item of the right part means the total flow into state j
The second item of the right part means the total flow out of state j.
And using the equations in line 1, we get the final form of probability flow equation in line 3.
The probability flow equation always is applied in such steps. First, we draw the state transition diagram, then we try to get the probability flow equation (using the idea: net=total in – total out). After that, we can get the performance evaluation and sometimes the formulas to typical systems.
For Queueing Theory Fundamentals, proceed to the next power point file. 1 2 3
Copyright by Yu-Chi Ho