On the rate-jitter performance of jitter-buffer in TDMoPSN: study using queueing models with a state-dependent service

Abstract

Time-division multiplexing over packet-switched network (TDMoPSN) is an intermediate phase of transition from current synchronous TDM to future all-optical converged network. TDMoPSN is exclusively used to transport interactive voice traffic transparently over a PSN (e.g., IP, MPLS or Ethernet). The goal of this paper is to reduce rate-jitter that is introduced into a stream of packets carrying TDM payload. We have proposed two online algorithms, algorithm-A and algorithm-B, to reduce the rate-jitter and shown analytically that the rate-jitter achieved by algorithm-A is strictly less than the rate-jitter of online algorithm proposed by Mansour et al. [1]. We have used three stochastic processes, namely Poisson, Markov-modulated Poisson process (MMPP) and an arrival process with Pareto-distributed inter-arrival times (see [2–4]) for modeling the arrival of TDM packets (say, IP packets with single or many TDM frames as payloads) at the destination. We undertook statistical analysis of the proposed algorithms by modeling the jitter-buffer as $M/\widetilde{D}/1/\, B_{on}$ and $MMPP/\widetilde{D}/1/B_{on}$ queues, to derive steady-state queue-length distribution, mean waiting time and distribution of inter-departure times. We also simulate the most realistic queueing model $Pareto/\widetilde{D}/1/B_{on}$ of our study and evaluated its performance with respect to the metrics: rate-jitter, mean waiting time, packet loss probability and steady-state queue-length distribution. Simulation results show that our proposed algorithms far outperforms the scheme proposed in [1]. We also present simulation results to verify the correctness of analytical queueing models. The algorithms proposed here are more general (for TDMoPSNs) and can be used to study TDMoIP, pseudowire, CES over metro Ethernet network (MEN), etc.

Appendices

Appendix 1: MMPP

MMPP can be viewed as a super-position of ’$M$’ independent Poisson processes, where switching among the processes is governed by an $M$-state CTMC, i.e., when the MMPP is in state $i$, arrivals occur according to a Poisson process with rate $\lambda _{i}$. The amount of time that the MMPP spends in state $j$ is exponentially distributed with mean $1/\mu _j$. If $N_{MMPP}(t)$ and $N_i(t)$ represent the counting process of MMPP and Poisson process with rate $\lambda _i$, at any arbitrary time $t$, then $N_{MMPP}(t)=N_i(t)$, whenever $J(t)=i$.

1.1 Derivation of average arrival rate $\lambda ^{*}$ of MMPP

The MMPP is characterized by the UMC $\{J(t),t\ge 0\}$, with transition rate matrix (infinitesimal generator matrix) $\mathbf {Q}^{\mathbf {*}}$ and arrival rate matrix $\varvec{{\varLambda }}$. For the four-state MMPP in Fig. 4, these matrices are given by,

$$\begin{aligned} \qquad \qquad \mathbf {Q}^{\mathbf {*}}=\begin{bmatrix} -\mu _{0}&\mu _{0}&0&0\\ p_{10}\mu _{1}&-\mu _{1}&(1-p_{10})\mu _{1}&0\\ 0&(1-p_{23})\mu _{2}&-\mu _{2}&p_{23}\mu _{2}\\ 0&0&\mu _{3}&-\mu _{3} \end{bmatrix} \end{aligned}$$

and $\varvec{{\varLambda }} = diag(\lambda _{0}, \lambda _{1}, \lambda _{2}, \lambda _{3})$. The steady-state probability vector $\varvec{{\varPhi }}=(\phi _{0},\phi _{1},\phi _{2},\phi _{3})$ of the UMC $\{J(t),t\ge 0\}$ can be obtained by solving the following equations:

$$\begin{aligned} \qquad \qquad \varvec{{\varPhi } Q^{*} = 0} \qquad \varvec{{\varPhi }\ e} = 1 \qquad \varvec{e}=(1,1,1,1)^{T} \end{aligned}$$

(47)

By solving Eq. (47), we get the steady-state probability vector $\varvec{{\varPhi }}$ as,

$$\begin{aligned} \varvec{{\varPhi }}= & {} \frac{1}{\theta } \Big (\mu _1\mu _2\mu _3 p_{10}(1-p_{23}), \mu _0\mu _2\mu _3(1-p_{23}), \\&\qquad \qquad \mu _0\mu _1\mu _3(1-p_{10}), \mu _0\mu _1\mu _2p_{23}(1-p_{10}) \Big ) \end{aligned}$$

where $\theta =\mu _0\mu _1(\mu _3+\mu _2p_{23})(1-p_{10})+\mu _2\mu _3(\mu _0+\mu _1p_{10})(1-p_{23})$.

Now, the average arrival rate $\lambda ^{*}$ of MMPP can be calculated as, $\lambda ^{*}=\varvec{{\varPhi } {\varLambda }\ e}$ and is given by,

$$\begin{aligned} \lambda ^*= & {} \frac{1}{\theta }[\lambda _0\mu _1\mu _2\mu _3p_{10}(1-p_{23})+\lambda _1\mu _0\mu _2\mu _3(1-p_{23}) \nonumber \\&\quad +\ \lambda _2\mu _0\mu _1\mu _3(1-p_{10})+\lambda _3\mu _0\mu _1\mu _2p_{23}(1-p_{10})]\nonumber \\ \end{aligned}$$

(48)

1.2 Calculation of blocks of $\mathbf {Q}$ matrix:

In this section, we would present an algorithm to compute the state transition probability transition matrix $\mathbf {Q}$ [Eq. (13)] of the bivariate EMC $\{(J_{k},Q^{D}_{k}),\, k\ge 0\}$. This means we need to compute the entries $\mathbf {A}(n,s_m)$ and $\mathbf {A}^{'}(n,s_1)$ of $\mathbf {Q}$ matrix. We closely follow the procedure given in [34], to compute the matrix $\mathbf {Q}$. The authors in [37, 38] presented a method to calculate the elements of blocks of matrix $\mathbf {Q}$, corresponding to the buffer occupancy EMC in a queue with general independent arrival process as input and a general distribution for service times, namely $(GI/PH/1\ \text {and}\ PH/G/1\ \text {queues})$. In [34], elements of $\mathbf {Q}$ matrix are computed for $N/G/1/K$ queue, where the input is Neuts process (N-process [35]), which is a more general form of MMPP.

To start with, we note that in [34], the infinitesimal generator matrix $\mathbf {Q}^{\mathbf {*}}$ is represented as, $\mathbf {Q}^\mathbf {*}=\mathbf {T}+\mathbf {T}^\mathbf {o}\mathbf {A}^\mathbf {o}$, all ($\mathbf {Q}^{\mathbf {*}}, \mathbf {T},\mathbf {T}^\mathbf {o}\ \text {and}\ \mathbf {A}^\mathbf {o}$) are $M\times M$ matrices. The columns of matrix $\mathbf {T}^\mathbf {o}$ are the vector $\bar{\mathbf {T}}^\mathbf {o}$ and $\mathbf {A}^\mathbf {o}\!=\!diag(\beta _0,\!\beta _1,\ldots ,\!\beta _{M-1})$ and $\bar{\varvec{\beta }}=(\beta _0,\beta _1,\ldots ,\beta _{M-1})$. For the four-state MMPP (Fig. 4), considered in this paper, these matrices are given by,

$$\begin{aligned}&\mathbf {T}=\begin{bmatrix} -\mu _{0}&\mu _{0}&0&0\\ p_{10}\mu _{1}&-\mu _{1}&(1-p_{10})\mu _{1}&0\\ 0&(1-p_{23})\mu _{2}&-\mu _{2}&p_{23}\mu _{2}\\ 0&0&0&-\mu _{3} \end{bmatrix}\\&\quad \bar{\mathbf {T}}^\mathbf {o}=\begin{bmatrix} 0 \\ 0 \\ 0 \\ \mu _{3}\\ \end{bmatrix} \\&\mathbf {A}^\mathbf {0}=diag(0,0,1,0), \qquad \bar{\varvec{\beta }}=[0\ 0\ 1\ 0] \end{aligned}$$

In [37, 38], it is proved that $\mathbf {A}(n,s_m)$ can be written as an infinite sum of product of $\gamma _l^{m}$ and $\mathbf {K}$ as below:

$$\begin{aligned} \mathbf {A}(n,s_m)=\overset{\infty }{\underset{l=n}{\sum }}\gamma _l^{m}\ \mathbf {K}^l_n \end{aligned}$$

(49)

where $\gamma _l^{m}$ is defined in Eq. (37) and $\eta =\underset{0\le i \le M-1}{\text{ max }}\Big (\big (-\mathbf {R}(0)\big )_{i,i}\Big )$.

Now, $\mathbf {K}_n^l$ is recursively defined as,

$$\begin{aligned} \begin{array}{lcl} \mathbf {K}_0^0 &{} = &{} \mathbf {I}, \qquad \mathbf {K}_0^{l+1}={\mathbf {C K}}_0^l \\ \mathbf {K}_n^0 &{} = &{} \mathbf {0}, \qquad \mathbf {K}_n^{l+1}={\mathbf {C K}}_n^l + \bar{\mathbf {C}}^0 \bar{\varvec{\beta } K}_{n-1}^l \end{array} \end{aligned}$$

where

$$\begin{aligned} \mathbf {C} = \eta ^{-1} \mathbf {T} + \mathbf {I}, \qquad \bar{\mathbf {C}}^0 = \eta ^{-1} \bar{\mathbf {T}}^0 \end{aligned}$$

In order to compute $\mathbf {A}(n,s_m)$ in Eq. (49) numerically, a truncation method is proposed in [38], which is an approximation, as given under:

In this method, a minimum value of index $N_0$ has to be found, such that $\overset{N_0}{\underset{l=0}{\sum }}\gamma _l^{m} \ge 1-\epsilon$, for any given small $\epsilon$ and $\forall s_m$. Then, $\mathbf {A}(n,s_m)$ is calculated as,

$$\begin{aligned} \mathbf {A}(n,s_m) = \overset{N_0}{\underset{l=n}{\sum }}\gamma _l^{m}\ \mathbf {K}_n^l + \Bigg (\overset{\infty }{\underset{l=N_0+1}{\sum }}\gamma _l^{m}\Bigg ) \mathbf {K}_n^{N_0} \end{aligned}$$

(50)

In order to compute first row of $\mathbf {Q}$, we identify, $\mathbf {A}^{'}(n,s_1) = {\mathbf {U A}}(n,s_1)$, where for MMPP input, $\mathbf {U}=(\varvec{{\varLambda }}-\mathbf {Q}^{\mathbf {*}})^{-1}\varvec{{\varLambda }}$ [36]. Therefore, $\mathbf {A}^{'}(n,s_1) = (\varvec{{\varLambda }}-\mathbf {Q}^{\mathbf {*}})^{-1}\varvec{{\varLambda }} \mathbf {A}(n,s_1)$. Similarly, the last column of $\mathbf {Q}$ matrix is computed using Eq. (14).

Appendix 2: Generalized Pareto-distributed inter-arrival process

The CDF of GPD is given by,

$$\begin{aligned} F(y)=1-\left( 1+\frac{\xi (y-\mu )}{\sigma }\right) ^{-\frac{1}{\xi }} \end{aligned}$$

(51)

Let,

$$\begin{aligned} \eta _{o}=\frac{\xi }{\sigma };\quad \lambda =\frac{1-\xi }{\sigma };\quad \mu =0 \end{aligned}$$

(52)

substitute these in Eq. (51) to get a new form for GPD.

$$\begin{aligned} F(t)=1-\left( 1+\eta _{o}t\right) ^{-\left( 1+\frac{\lambda }{\eta _{o}}\right) } \end{aligned}$$

(53)

where $\lambda$ is the average arrival rate, i.e., $X_{a}=1/\lambda$ is the average IAT of GPD.

The variance $v$ of the GPD can be shown as,

$$\begin{aligned} v=\frac{\sigma ^{2}}{(1-\xi )^{2}(1-2\xi )} \end{aligned}$$

(54)

Variance is finite if $\xi <0.5$. Now, by using Eqs. (52) and (54), we can derive the parameters of the GPD. They are given by,

$$\begin{aligned} \xi = \frac{v \lambda ^2 - 1}{2 v \lambda ^2}; \sigma = \frac{v \lambda ^2 + 1}{2 v \lambda ^3}; \eta _{o} = \frac{\xi }{\sigma } = \lambda \left( \frac{v \lambda ^2 - 1}{v \lambda ^2 + 1} \right) \end{aligned}$$

(55)

The parameters in Eq. (55) are used to generate Pareto-distributed random numbers by using the inverse CDF method.