Exponential Model of Token Bucket System

There is a flow of packets entering the TBS. The Token Bucket System is generating tokens (marks) and then it marks each packet with one of them. Only marked packets will go through the network. For practical reasons we can assume a limited bucket with depth “n”. In case the bucket is empty (there are no tokens), TBS cannot mark any packet, which is lost then. Let Plst be probabilities of this random phenomenon and we will call it “probability of loosing packet”.


Steady-state Analysis
The Token Bucket System is related to VoIP problems. We have formed the steady-state analysis of VoIP under the Token Bucket Control. Our main problems will be to compute the probability characteristics of TBS and to find relation between probability of a loosing packet and the bucket depth. At first we will analyze the work of TBS in general (Fig. 1).
There is a flow of packets entering the TBS. The Token Bucket System is generating tokens (marks) and then it marks each packet with one of them. Only marked packets will go through the network. For practical reasons we can assume a limited bucket with depth "n". In case the bucket is empty (there are no tokens), TBS cannot mark any packet, which is lost then. Let P lst be probabilities of this random phenomenon and we will call it "probability of loosing packet".
We will deal with simple Exponential model of On-Off Source of human speech. This Source has two states. The state On is period of "speech" and the Off state is period of "silence". Between these states, the source switches randomly. Talk-spurt duration is modelled by variant T 1 and pause duration is modelled by variant T 2 . The values of distribution parameters were gained from [1].
When the Source is in On state it starts generating a flow of packets, which represents human speech. The packet rate in usual VoIP systems (using G.729A) is 50 p/s. We will assume that the flow of packets is modelled by Poisson process N P (t) with rate ϭ 0.05p/ms .
In general there can be any token rate. Usually it is the same as an average number of packets entering the TBS. Let P ON and P OFF be probabilities that the Source is in state On or Off. The flow of tokens will be modelled by Poisson process N T (t) with rate ϭ . P ON ϩ 0 . P OFF .
We will construct a transition diagram of Markov model of On-Off Source ( Let Q 0 be a rate matrix for this chain. We compute state probabilities of a steady-state chain by balance equations: We compute probabilities and token rate for real parameters of VoIP: In the Token Bucket System we have three elementary random phenomena, entering the packet, generating token and switching between On-Off states in Source. We have assumed that each of them is Poisson process, and because of that we can model this TBS by Markov chain. If there is an arriving packet and the bucket is empty, we cannot mark it and this packet is lost. Probability of loosing packet was named P lst . We will construct a transition diagram of the whole system (Fig. 3). Component columns of diagram refers to a number of tokens in the token bucket. We have assigned this as "k-level" for k ϭ 0, …, n. For easy reading loops in the transition diagram are omitted: Let X k (Y k ) be a probability that in the TBS are k tokens and Source is in state On (Off). Let P k be a probability of k-level. We see that P k ϭ X k ϩ Y k and P lst ϭ X 0 . Taking from the Theory of Markov Chains [2] we can write the following equations for state probabilities of the steady-state chain: We will solve the steady-state equations QЈp ϭ 0, where p ϭ (Y n , X n , …, Y k , X k , … Y 0 , X 0 ) and matrix QЈ is formed as: We modified the rate matrix QЈ to the equivalent triangular matrix and we used a normalization condition for state probabilities We gained the recurrent formulas for state probabilities: We can reduce formulas for Y k and Y n to:

TBS with real parameters of VoIP
To deduce explicit formula for probability of loosing packet P lst ϭ P lst (n) ϭ X 0 is difficult and impracticle, but there is no problem to compute the values of P lst for real parameters of our TBS with Exponential On-Off Source: for k ϭ 1, …, n, Y k ϭ 0.97002 и Y kϪ1 ϩ 0.07862 и X kϪ1 for k ϭ 1, …, n, For example, the reader can see the difference between n ϭ 4 and n ϭ 5: For real use its enough to have the bucket depth n ϭ 1, …, 10. Now we will increase the bucket depth n and calculate characteristics of models: P lst (n) -probability of loosing packet or probability of empty bucket in time of arriving packet . P lst (n) -average number of lost packets P n -probability of full bucket (bucket will refuse tokens) EK -average bucket depth Ꭽ ϭ ᎏ E n K ᎏ -token bucket usage

Relation between Probability of loosing packet and the Token Bucket depth
The most interesting characteristic is probability of loosing packet P lst (n) . Its values will be approximated by regression functions Y I (n): by the least squares method. We will measure the quality of approximation by square of sum residuals: