The Optical Buffer Queueing System

This section introduces the developed optical buffer queueing model which approximates the be- haviour of the contention resolution scheme. Consider that the optical buffers consist of L Fibre De- lay Lines (FDLs), and each fibre contains S outgoing channels. As indicated in the previous section, the optical buffering network is decomposed into S parallel Markov-modulated finite-server queues, and the input process to the queue is represented by R two-state MMPPs (

(

Q_j,

(

Λj), j = 0, . . . , R−1, see Figure 3.3. Note that the resource scheduling mechanism in the proposed switch relies on the PreRes scheme which reserves the switch and FDL resources before the packets enter into the switch. This strategy determines that the two-state MMPP traffic (

(

Q_j,

(

Λj), for 1 ≤ j ≤ R − 1, which is about to traverse through the optical buffer for the (j + 1)-th time, goes though the buffer- ing queue without any traffic loss, and turns into the (j + 1)-th input retrial flow directly. That means, in the buffer queueing analysis, the R MMPP traffic streams (

(

Q_j,

(

Λj), j = 0, . . . , R − 1, will compete and occupy the resources, but the MMPPs (

(

Q_j,

(

Λj), j = 1, . . . , R − 1, have higher priority, and only the MMPP (

(

Q₀,

(

The optical buffers utilised are assumed to be of degenerate type, that is, the discrete-time delays introduced by FDLs are integer multiples of the delay granularity D, i.e. D, 2D, . . . , LD. It is ob- vious that for small values of D, the time resolution of the FDL buffers is low, thus the behaviours of the optical buffers can be well modelled by a queueing system with L servers operating in paral- lel [12, 17]. When D is large, the time resolution of the FDL buffers is high, and in this case each fibre is more likely to function as an individual queue and carry the incoming traffic independently, so the buffers can be modelled as a cascaded chain of L distinct queues [22]. Depending on the FDL delay granularity, two different Markov-modulated finite-server queueing models are devel- oped: (a) an MMPP/M/L queue, as shown in Figure 3.4(a), and (b) a chain of L MMPP/M/1 queues with cascading overflows, as shown in Figure 3.4(b). The MMPP/M/L queue is expected to approx- imate the performance of the FDLs with small D, whereas the cascaded model of L MMPP/M/1 queues is expected to quantify the optical buffers with large D. As shown in Figure 3.4(a), the MMPP/M/L model is a L-server queue receiving MMPP-based arrival process which includes R two-state MMPPs (

(

Q_j,

(

Λj), j = 0, . . . , R − 1. The arrival process (

( Q,Λ) is computed as,( ( Q = ( Q₀⊕ ( Q₁⊕ · · · ⊕ ( Q_R−1, Λ =( ( Λ0⊕ ( Λ1⊕ · · · ⊕ ( ΛR−1. (3.23)

The MMPP/M/L queueing model can be approximated by a L + 1-state Markov process where the infinitesimal generatorQ is expressed as,ˇˇ

ˇ ˇ Q = 0 1 .. . L         ( Q −Λ( Λ( I Q −( Λ −I( Λ( . .. . .. . .. LI Q −LI(         . (3.24)

When the Markov process is in the state i, i ∈ {0, . . . , L − 1}, the arrival traffic is processed and departs from the system, but if the process is in the state L, which means all of the L servers are busy, the arrival traffic overflows from the queue. The MMPP departing processes, which are quantified by two-state MMPPs (Q´_j, Λ´_j), j = 0, . . . , R − 1, will loop back to the switch queueing system as part of the arrival process. Conversely, the traffic overflowing from the queue will be abandoned and considered lost from the network. Differently, in the cascaded model of L MMPP/M/1 queues, the arrival process to the first MMPP/M/1 is MMPP (Q,( Λ). The overflowing process, modelled( as MMPP, forms the arrival process to the next MMPP/M/1, and so on down the chain, as shown in Figure 3.4(b). In the L-th MMPP/M/1 queue, the overflow traffic from the queue is discarded.

(a) An MMPP/M/L queue

(b) A cascaded model of L MMPP/M/1 queues

Figure 3.4: The buffer queueing model consists of S parallel Markov-modulated finite-server queues. Two different representations are developed to characterise the Markov-modulated finite- server queue: (a) a single MMPP/M/L model and (b) a cascaded model of L MMPP/M/1 queues.

Obviously, the departing process from the cascaded model is the superposition of L individual MMPP departing streams. Note that these individual streams are not identical. Further, the multi- state departing processes are approximated by two-state MMPPs (Q´_j, Λ´_j), j = 0, . . . , R − 1, so as to simplify the analysis. It is important to point out that in both buffer models, for j = 1, . . . , R − 1, the MMPP (Q´_j, Λ´

j) and the MMPP (

(

Q_j,

(

Λj) are the same process, due to the PreRes scheme. Therefore, the overall departing process from the buffer queueing model is composed of S identical two-state MMPPs (Q´_j, Λ´_j), which is then reduced to a two-state MMPP ( ¨Qj, ¨Λj), j = 0, . . . , R−1. The MMPP ( ¨Qj, ¨Λj) is uniformly distributed to N K switch models as arrival process (Qj+1, Λj+1), j = 0, . . . , R − 1, which is MMPP (Qi, Λi), i = 1, . . . , R.

An important feature of the cascaded model is that it keeps track of the individual traffic flows, from and to, each FDL, thus allowing for the estimation of the mean occupancy of each FDL. Given the

occupancy measures M = {M (k)|1 ≤ k ≤ L}, the overall end-to-end delay T is defined by T = D L X k=1 kM (k). (3.25)

The equation reveals that a linear dependency exists between the mean communication delay and the FDL base delay D.

To validate the performance of the analytical model, a detailed evaluation of the proposed analytical framework has been performed in the next section, and extensive comparisons against simulation results have been carried out.