The PRNN-based Prediction

First, we define notations in the following:

cycle n (of ONUi) : the time difference between the OLT first beginning to receive the nth REPORT message from ONUi and the next beginning to receive the (n+1)st REPORT message from the same ONUi;

0_,i(n)

L : the reported occupancy of queue Q in bytes at cycle n; _0,i

1_,i(n)

L : the reported occupancy of queue Q in bytes at cycle n; _1,i

2_,i(n)

L : the reported occupancy of queue Q in bytes at cycle n; _2,i

dp ,i(n)

L : the total amount of video packets (in bytes at cycle n) which will be dropped at the end of the next cycle if they are not transmitted at the next cycle because their delay time will violate the delay constraint;

*

T : the delay constraint of video packets in seconds; d d ,i(n)

L : the total amount of video packets (in bytes at cycle n) which should be transmitted at the next timeslot in order to sustain the requirement of packet dropping probability;

*

P : the dropping probability bound of video packet; d w,i(n)

L : the total amount of data packets (in bytes at cycle n) whose waiting time is larger than a waiting bound;

*

T : the waiting bound of data packets in seconds; w m,i(n)

G : the granted bandwidth in bytes for Q_m,i at cycle n;

m,i(n)

L : the reported occupancy of queues Q_m,i at cycle n;

m,i(n)

E~ : the amount of estimated new arrival packets for Q_m,i in bytesduring cycle n;

m,i(n)

P : the predicted occupancy of Q_m,i in bytes during cycle n;

i(n)

T : the cycle time of EPON in seconds at cycle n for ONUi; (n)

m,i

λ~ : the estimated packet arrival rate in bytes/sec. during cycle n;

m,i(n)

A : the amount of actual new arrival packets at Q_m,i in bytes during cycle n;

m,i(n)

λ : the actual packet arrival rate in bytes/sec. during cycle n.

The proposed PRNN-based predictive QoS-promoted dynamic bandwidth allocation (PRNN-based predictive Q-DBA method) assumes that ONUi, 1 ≤ i ≤ M, sends report message including six sorts of information of queues,L_0,i(n),L_1,i(n),L_2,i(n),L_{dp ,i}(n),L_{d ,i}(n), andL_w,i(n). TheL_0,i(n),L_1,i(n), andL_2,i(n)denote the reported occupancy of queuesQ ,_0,i Q , _1,i andQ in bytes at cycle n, respectively. The_2,i L_{dp ,i}(n)denotes the total amount of video packets (in bytes at cycle n) which will be dropped at the end of the next cycle if they are not transmitted at the next cycle because their delay time will violate the delay constraint of video packet, T . The_d^* L_{d ,i}(n)denotes the total amount of video packets (in bytes at cycle n) which should be transmitted at the next timeslot in order to sustain the requirement of packet dropping probability of video packet, P . The_d^* L_w,i(n)denotes the total amount of data packets (in bytes at cycle n) whose waiting time is larger than a waiting bound, T . _w^*

Figure 3.4: Upstream timeslot assignments

Figure 3.4 shows the upstream timeslot assignment between the OLT and the ONU. “A cycle n of ONUi” is defined as the time difference between the OLT first beginning to receive the nth REPORT message from ONUi and the next beginning to receive the (n+1)st REPORT message from the same ONUi (As depicted in Figure 3.4, T5-T1 is the cycle n-1 for ONU¹).

When the ONU receives the GATE message from the OLT, it is ready for ONU to transmit packets at its timeslot. For example, as shown in Figure 3.4, upon receiving G_m,1(n-1) , m∈0, 1, 2, the granted bandwidth in bytes for Q_m,1 with service type m at cycle n-1, ONU¹ starts to transmit packets at t1 (assigned by OLT), the ONU¹ will end its transmission at t2 and give a report message to OLT with the reported queue occupancy. In the meantime between T1 and T5, the ONU¹ will keep receiving packets from the user.

Assume that the OLT is at the present cycle n-1(start at T1, as depicted in Figure 3.4). In order to make a better utilization of bandwidth, we need to consider the new arrival packets at cycle n-1 for ONUi.That is, we make a prediction of the amount of ONUi’s new arrival packets at cycle n-1.

Denote ~_m,i(n-1)

E to be the amount of estimated new arrival packets of Q_m,i in bytes at cycle n-1 and P_m,i(n)to be the predicted occupancy of Q_m,i in bytes at cycle n. P_m,i(n) can be Denote T_i(n-1) to be the cycle time of EPON in seconds at cycle n-1 for ONUi

and m,i(n-1)

~λ to be the estimated packet arrival rate in bytes/sec. during cycle n-1.

The ~_m,i(n-1)

E can be calculated as follows

, ,

~ ~

(n-1) (n-1) * (n-1)

m i m i i

E =λ T . (2) Through the analysis of the relationship between the reported queue occupancy and the granted bandwidth, we could find the way to get the amount of actual new arrival packets at Q . For example, let_m,i A_m,i(n-2) to be the amount of actual new arrival packets at Q_m,i in bytes at cycle n-2. At cycle n-1, the OLT can obtain the amount of actual new arrival packets

(n-2)

Am,i according to the reported queue occupancy of Q ,_m,i L_m,i(n-1), L_m,i(n-2)and the granted bandwidthG_m,i(n-1). If the reported queue occupancy L_m,i(n-1) is greater than zero, then the amount of actual new arrival packets A_m,i(n-2) can be calculated from

(n-1)

Gm,i , L_m,i(n-1), and L_m,i(n-2). If the reported queue occupancy equals to zero, assume that the actual new arrival packets arrive in uniform distribution, then the A_m,i(n-2) can be obtained as follows

Denote λ_m,i(n-2) to be the actual packet arrival rate at cycle n-2, it can be calculated as follows As mentioned earlier, in order to make a better utilization of bandwidth, we need to consider the mount of new arrival packets to the ONU between each ONU’s report time. In this thesis, we estimate the mount of new arrival packets using the PRNN predictor. As depicted in Figure 3.6, let λ_m,i(n-2) to be the input of PRNN predictor, then the estimated packet arrival rate m,i(n-1)

λ~ can be obtained from the output of PRNN predictor. For example, as depicted in Figure 3.4, the OLT can make a prediction of the new packet arrival rate m,1(n-1)

λ~ at T2, then m,2(n-1)

~λ at T3, and so forth. After the OLT has collected all the predicted data for ONUi at cycle n-1 (at T4), the proposed DBA mechanism allocates the available bandwidth to each ONU. The details of proposed DBA mechanism will be discussed in next section and the description of PRNN predictor will be introduced in the following.

Least mean square (LMS) and recursive least squares (RLS) are two linear adaptive structures for the prediction of signals. LMS is firstly introduced by Widrow and Hoff in 1959, the strength of the LMS resides on its simplicity and easy processing. Afterward there are some improved algorithms for LMS, such as normalized least mean square (NLMS), fast least mean square (FLMS), discrete cosine transform - least mean square (DCT - LMS) [10]. RLS is a representative of the other prediction algorithms, it is based on the method of the least squares on the theory of Kalman filters. The main difference between LMS and RLS is the

inherent statistical conception. In RLS algorithm, we work with time-based averages calculated from different samples of the same random process, but on the contrary, in LMS algorithm, averaging involves values acquire from specific time but from different realization of one random process. Because of the calculation of time average requires more instructions, so the computational complexity of RLS-based is one order higher than that of LMS-based methods. But from the point of view of the performance of convergence rate, the RLS-based algorithms are several times speedy than LMS-based algorithms.

In 1995, Haykin and Li [11] presented a nonlinear predictor based on the pipelined recurrent neural network (PRNN). The PRNN is believed to outperform the LMS and RLS in the prediction of nonlinear and non-stationary signals. The PRNN is composed of a number of small recurrent neutral networks (RNN’s) but keep its relatively low computational complexity. A real-time recurrent learning algorithm (RTRL) [12] is used by Haykin and Li for the training of the PRNN and it helps to predict the nonlinear and non-stationary signals more precisely and accurately.

The PRNN predictor is a modular neural network, and consists of a certain number r of RNN’s as its module, each module is composed of N neurons. The PRNN predictor has good nonlinear prediction capability and fast convergent time. A fully connected RNN structure, which has N neutrons and p＋q＋N input nodes, is shown in Fig.3.5. The predicted data

(n+1)

m,i

λ~ is expressed as follows (For simplicity, the lower subscript is omitted.)

(n+1) ( (n) (n-p+1) 1 (n) (n-q+1) 2(n-1) _N(n-1))

~ ~ ~ ~ ~

H ,..., ; ; ,..., ; y ,..., y

λ = λ λ λ λ , (5)

where ( )H ⋅ is an unknown nonlinear function. The j th neutron first calculate a weighted sum, denoted by ( )v n , and is given by equation (6) _j where wij denotes the weight of the connection from the ith input node to the jth neuron, ui(n) is the ith input node. The output yj(n) is the transformation of the vj(n) by a sigmoidal activation function, which is given by

( ( ) )

( 1) ( ) C n( )

w n w n ,

η^∂ w

+ = −

∂ (8) where η is a fixed learning rate parameter, C(n) is the cost function defined as

1 2

As shown in Fig.3.6, the PRNN is a pipelined structure of the NARMA-based RNN predictor. All the modules of the PRNN are designed to have exactly the same synaptic weight matrix W. The predicted data m,i(n-1)

~λ can be obtained from the first neutron of the first module.

…

Figure 3.5: Fully Connected Structure of the RNN

~ λ(n+1)

Figure 3.6: Pipelined Recurrent Neural Network (PRNN)

~ λ(n-1)