## Abstract

We provide here a new loss model for an optical hybrid switch that can function as an optical burst switch and/or optical circuit switch. Our model is general as it considers an implementation whereby some of the circuits have preemptive priority over bursts and others are allowed to queue their reservations. We first present an analysis based on a 3-dimension state-space Markov chain that provides exact results for the blocking probabilities of bursts and circuits, the proportion of circuits that are delayed and the mean delay of the circuits that are delayed. Because it is difficult to exactly compute the blocking probability in realistic scenarios with a large number of wavelengths, we derive computationally a scalable and accurate approximations based on reducing the 3-dimension state space into a single dimension. These scalable approximations that can produce performance results in a fraction of a second can readily enable switch dimensioning. Extensive numerical results are presented to demonstrate the accuracy and the use of the new approximations.

© 2006 Optical Society of America

## 1. Introduction

The recent interest in optical burst switching (OBS) [6, 7, 8, 9, 10, 11, 12] has given rise to the concept of optical hybrid (circuit/burst) switching (OHS) [1, 2, 3, 4, 5] which supports both optical circuit switching (OCS) and OBS. OHS provides a solution that combines the benefits of OBS and OCS and may provide evolutionary path beyond OCS. Connections that will use the OCS part can enjoy its predictable and reliable performance while optical bursts can be used to improve efficiency in resource utilization and for more flexible connectivity.

We consider here an extension of our OHS model with circuit queued reservations [4]. The model in [4] considers an implementation where circuits are allowed to wait until the traffic that uses its intended path clears out so that its end-to-end path can be set up. This is equivalent to a situation where advanced reservation is made for a circuit when it cannot be admitted right away. The extension here includes a premium circuit switching class that can preempt bursts in progress. In particular, when a request for a circuit of this premium class arrives at an OHS node, and all output wavelengths in the direction required by that premium circuit are occupied - at least one of them by a burst - then the premium circuit will preempt a burst (or burst reservation if the burst has not arrived yet). If all the relevant output wavelengths are busy by circuits, the premium request cannot be set up immediately and may be queued for future set-up. Circuits, premium or non-premium, may be queued if all relevant access are busy, however, if too many circuits are already queued, some users may choose not to queue, or that some users will not be allowed to queue if the number in the circuit queue is above a certain threshold. This allows classifications between circuits. Notice that the queue threshold can be set to zero for some users. Such users may not be allowed to queue their circuit requests. All in all, we consider here a model of an OHS node that supports OBS traffic (bursts) as well as several priority classes of OCS traffic (circuits) that do not have preemptive priority over bursts and a premium class of circuit traffic that has preemptive priority over bursts. This potentially enables provision of quality of service (QoS) classifications to circuit traffic while bursts may use the scraps for better utilization.

By OCS we mean all cases where connection between edge routers is set up Edge-to-Edge and capacity is exclusively available for the entire duration of the connection. These include scenarios where capacity is permanently or semi-permanently available - the so-called Static OCS. By OCS, we also include cases where connections are set up and taken down frequently (Dynamic OCS). Our OCS concept also includes the so-called OBS with acknowledgement [13], or the similar earlier proposal called Wavelength Routed OBS [14]. The fact that OCS includes such a wide range of options justified the need for differentiation to many classes. For example, a circuit set up for a leased line between major cities for a long period of time may be able to wait longer than a short circuit set up as OBS with acknowledgement.

On the other hand, by OBS we include all methods of optically transmitting and routing bursts of data using one way reservation, but without the ability to buffer the bursts optically. In other words, packet data are aggregated in edge routers in large buffers from where they are transmitted optically to other edge routers through the core network without being buffered on their way in the core network. While data is normally not lost inside the core optical network under OCS as capacity is exclusively guaranteed for the connection, under OBS, loss may occur if too many bursts arrive from many input wavelengths to be forwarded to the same output link, but there are not enough suitable output wavelengths there to accommodate them. In such a case, a burst is dumped.

Although, for our problem, a 3-dimension state-space Markov chain can lead to exact performance results, such analysis is not scalable for the system with hundreds of wavelengths. Furthermore, for such a system, computer simulation is also not scalable as it takes a very long time to visit the large number of states, a sufficient number of times to obtain accurate results. The time required is especially longer if we are interested in the probability of a rare event such as blocking.We remind the reader of numerous studies published on electronic hybrid switching over the last 30 years (See [16, 17] and references therein). However, due to the “curse of dimensionality”, no accurate approximations of the type proposed in these papers have been provided.

An alternative model without allowing circuits to wait is considered in [3, 15]. In [5], an OHS with circuit queued reservations was considered where all circuits have preemptive priority over bursts. The additional flexibility and generality considered here of allowing only part of the circuit preemptive priority over bursts imposes a further analytical challenge.

Our contributions in this paper are scalable and accurate approximations for the blocking probability of circuits and bursts, proportion of circuits that are delayed and the mean delay of the circuits that are delayed. The approximation is based on simplifying a 3-dimensional Markov chain into a single dimensional Markov chain which characterizes sufficiently accurately the many kinds of traffic we consider: transmitted bursts, dumped bursts, circuits with preemptive-priority over bursts (in service and in the queue) and circuits without preemptivepriority over bursts (in service and in the queue). Because it is difficult to directly characterize all these interacting traffic types, so that they are merged into a single measure in one step, the approximation is based on a fixed point iterative procedure. The scalable approximations provided in this paper that can produce performance results in a fraction of a second can readily enable switch dimensioning, meaning that for any given traffic loading we can compute quickly the performance measures so we can choose the right switch capacity such that required quality of service is obtained.

## 2. The model

Our model comprises an output link of an Optical Cross Connect (OXC) and all the input wavelengths where traffic is transmitted towards the output link under consideration. Our output link is assumed to have *F* optical fibers, each of which carries *W* wavelengths. In the case of full wavelength conversion, this output link is considered to have *C*=*FW* wavelength channels. Otherwise, a burst/circuit that arrives on a certain input wavelength must use the same wavelength at the output. In this case, the output wavelength is considered to have only *F* wavelength channels. Let *K* be the number of wavelengths in our output link. Accordingly, *K* takes the values of *C* or *F* for the cases of with or without wavelength conversion, respectively.

Let *M* be the number of input wavelengths from where traffic flows are arriving towards our output link. As in [3], we assume full wavelength conversion capabilities in the switching fabric and that the switching fabric is strictly non-blocking. Under this assumption, if *M*≤*K* then no loss will occur. Therefore, computing blocking probability is only relevant for the case *K*<*M*. Figure 1 shows the optical hybrid switching transport network architecture and Fig. 2 shows the optical hybrid switch architecture.

For simplicity, throughout the paper, we ignore effect of OBS offset time [25]. This assumption can be justified for several OBS proposals. The Optical Coarse Packet Switching proposal [21] eliminates the offset altogether. The Dual-header OBS [22] insures a constant offset so a model based on zero offset assumption will give accurate blocking results but the delay results will need to be increased by a constant. Just-in-time (JIT) OBS [23] assumes that an immediate reservation is made by the header so all is needed is to adjust the burst length (input parameter) to apply the results of this paper. However, in other OBS implementations, offsets may have effect on blocking probability that are not easily tractable [24, 28]. Nevertheless, in any case, based on the particular method used, one may adjust certain input parameters or the results of this study to accurately evaluate performance measures.

We assume that a burst and a circuit are transmitted on a wavelength for an exponentially distributed period of time with means 1/*µ*_{b}
and 1/*µ*_{c}
, respectively. For each input wavelength, we assume that the traffic behaves as an on-off process. This on-off process can be viewed as a process with two alternating states – on and off. A period of time used to transmit a single burst or allocated for a circuit on an input wavelength is called an *on* period, and the time between consecutive on periods on that input wavelength is called an *off* period. The off period on an input wavelength is assumed to be exponentially distributed with mean 1/*λ*. This assumption of exponential times is not as limiting as it may seem. It is well known that Engset formula [29] is insensitive to on and off distribution [30]. The model we consider here has exponential times, but it has been shown that the burst blocking probability is not too sensitive to the distribution of the on and off periods [3, 10, 15, 31], where there are no circuit queued reservations. Although the sensitivity of the blocking probability to the on and the off distribution is likely to increase with the maximal number of queued circuits (denoted as *N*), it should not increase too much since *N* is designed to be small.

Upon termination of an off period, an on period associated with a burst transmission will commence with probability *p*_{b}
, or a circuit transmission will commence with probability *p*_{c}
=1-*p*_{b}
. For a circuit, with probability *p*_{pc}
, it has preemptive priority over burst (we call it premium circuit) while with probability *p*_{nc}
=1-*p*_{pc}
it does not have preemptive priority over burst (we call it non-premium circuit). Note that the probability *p*_{pc}
that represents the proportion of premium circuits out of all circuits is henceforth referred to a Premium Circuit Proportion (*PCP*). Define *λ*_{c}
=*λ*_{pc}
, *λ*_{pc}
=*λ*_{c}*p*_{pc}
, *λ*_{nc}
=*λ*_{c}*p*_{nc}
and *λ*_{b}
=*λ*_{pb}
. In addition, we assume that if all K output wavelengths are busy, *c*(*n*) (0≤*c*(*n*)≤1,*n*=0,1,2,…,*N*,*N*≤*M*-*K*) proportion of the circuit arrivals are willing to wait until no more than n items (circuits and/or bursts) complete their transmission. For example, *c*(0) represents the proportion of the circuit arrivals which are not willing to wait any time and just leave the system immediately if the system is busy. Thus, their arrival rate is *c*(0)*λ*_{c}
.

## 3. Exact solution

Let *π*_{i,j,k}
(*i,j,k*≥0, *i*≤*K, k*≤*M*-*K*, *j*≤*K*+*N*) be the probability that *i* input wavelengths are used for bursts transmission, *j* are for transmitting and for waiting circuits, and *k* are for dumping blocked bursts. The number of idle input wavelengths is given by *M*-*i*-*j*-*k*. The *π*_{i,j,k}
values are obtained by the following steady state equations. For *i*+*j*<*K*,

$$={\pi}_{i,j,k+1}\left(k+1\right){\mu}_{b}+{\pi}_{i,j+1,k}\left(j+1\right){\mu}_{c}$$

$$+{\pi}_{i,j-1,k}\left(M-\left(i+j-1+k\right)\right){\lambda}_{c}$$

$$+{\pi}_{i-1,j,k}\left(M-\left(i-1+j+k\right)\right){\lambda}_{b}$$

$$+{\pi}_{i+1,j,k}\left(i+1\right){\mu}_{b},$$

and for *i*+*j*=*K*,

$$={\pi}_{i,j-1,k}\left(M-K+1-k\right){\lambda}_{c}$$

$$+{\pi}_{i-1,j,k}\left(M-K+1-k\right){\lambda}_{b}$$

$$+{\pi}_{i,j,k-1}\left(M-K-k+1\right){\lambda}_{b}+{\pi}_{i,j,k+1}\left(k+1\right){\mu}_{b}$$

$$+{\pi}_{i,j+1,k}\left(j{\mu}_{c}\right)$$

$$+\phantom{\rule{.2em}{0ex}}+{\pi}_{i+1,j,k}\left(i+1\right){\mu}_{b}+{\pi}_{i+1,j-1,k-1}\left(M-K-k+1\right){\lambda}_{\mathit{pc}}.$$

and for *K*<*i*+*j*≤*K*+*N*,

$$={\pi}_{i,j-1,k}\left(M-i-j-k+1\right){a}_{j-1}\left({\lambda}_{\mathit{nc}}+{\delta}_{i}{\lambda}_{\mathit{pc}}\right)$$

$$+{\pi}_{i+1,j-1,k-1}\left(M-i-j-k+1\right){a}_{j-1}{\lambda}_{\mathit{pc}}$$

$$+{\pi}_{i,j,k-1}\left(M-i-j-k+1\right){\lambda}_{b}+{\pi}_{i,j+1,k}\left(K-i\right){\mu}_{c}$$

$$+{\pi}_{i,j,k+1}\left(k+1\right){\mu}_{b}+{\pi}_{i+1,j,k}\left(i+1\right){\mu}_{b}$$

where *δ*_{i}
=1 if *i*=0 and *δ*_{i}
=0, otherwise, and

${a}_{j}=\sum _{n=i+j-K+1}^{N}c\left(n\right).$ .

Introducing the normalization equation

$\sum _{i,j,k}{\pi}_{i,j,k}=1$

gives rise to a set of equations, which can be used to obtain the *π*_{i,j,k}
probabilities.

The total load offered by premium circuits is given by

${O}_{\mathit{pc}}=\sum _{i,j,k}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{pc}}}{{\mu}_{c}}\right){\pi}_{i,j,k}.$ .

The total load offered by non-premium circuits is given by

${O}_{\mathit{nc}}=\sum _{i,j,k}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{nc}}}{{\mu}_{c}}\right){\pi}_{i,j,k}.$ .

The total load carried by premium circuits is given by

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\sum _{i=0,K\le i+j<K+N,k}{a}_{j}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{pc}}}{{\mu}_{c}}\right){\pi}_{i,j,k}.$$

The total load carried by non-premium circuits is given by

${C}_{\mathit{nc}}=\sum _{i+j<K,k}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{nc}}}{{\mu}_{c}}\right){\pi}_{i,j,k}+\sum _{K\le i+j<K+N,k}{a}_{j}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{nc}}}{{\mu}_{c}}\right){\pi}_{i,j,k}.$ .

The premium circuit blocking probability *B*_{pc}
is given by

${B}_{\mathit{pc}}=1-\frac{{C}_{\mathit{pc}}}{{O}_{\mathit{pc}}},$ ,

and, the non-premium circuit blocking probability *B*_{nc}
is given by

${B}_{\mathit{nc}}=1-\frac{{C}_{\mathit{nc}}}{{O}_{\mathit{nc}}}.$ .

Therefore, the average circuit blocking probability *B*_{c}
is given by

${B}_{c}=1-\frac{{C}_{\mathit{pc}}+{C}_{\mathit{nc}}}{{O}_{\mathit{pc}}+{O}_{\mathit{nc}}}.$ .

The total load offered by bursts is

${O}_{b}=\sum _{i,j,k}\left(M-i-j-k\right)\left(\frac{{\lambda}_{b}}{{\mu}_{b}}\right){\pi}_{i,j,k},$ ,

and the total load carried by bursts is

${C}_{b}=\sum _{i,j,k}i{\pi}_{i,j,k}.$ .

Alternatively, *C*_{b}
can be calculated by the burst traffic accepted by the system minus the lost burst traffic due to circuit preemption, as follows

${C}_{b}=\sum _{i+j<K,k}\left(M-i-j-k\right)\left(\frac{{\lambda}_{b}}{{\mu}_{b}}\right){\pi}_{i,j,k}-\sum _{i>0,K\le i+j\le K+N,k}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{pc}}}{{\mu}_{b}}\right){\pi}_{i,j,k}.$ .

Thus, the burst blocking probability *B*_{b}
is given by

${B}_{b}=1-\frac{{C}_{b}}{{O}_{b}}.$ .

The carried load of circuits in the queue is

${C}_{q}=\sum _{\forall k,K\le i+j<K+N}{a}_{j}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{nc}}}{{\mu}_{c}}\right){\pi}_{i,j,k}+\sum _{\forall k,i=0,K\le i+j<K+N}{a}_{j}\left(M-i-j-k\right)\left(\frac{{\lambda}_{\mathit{pc}}}{{\mu}_{c}}\right){\pi}_{i,j,k}.$ .

The proportion of circuits that are delayed, denoted as *D*_{p}
, is

${D}_{p}=\frac{{C}_{\mathit{qc}}}{{C}_{c}}.$ .

The average number of circuits, denoted as *N*_{c}
, in the system is given by

${N}_{c}=\sum _{\forall i,j,k}j{\pi}_{i,j,k}.$ .

By Little’s formula, the mean circuit delay is given by

${D}_{c}=\frac{{N}_{c}}{{C}_{c}{\mu}_{c}}.$ .

The mean circuit queueing delay is given by

The mean queueing delay for a delayed circuit, denoted as *Q*_{qc}
, is given by

Solving Eqs. (1), (2) and (3) is not scalable for large *K* and *M*, so scalable approximations are derived next.

## 4. Approximation

The approximation is based on reducing the above described three dimensional Markov chain into a single dimensional Markov chain which characterizes sufficiently accurately the many types of traffic that we have here: transmitted bursts, dumped bursts, circuits with preemptivepriority over bursts (in service and in the queue) and circuits without preemptive-priority over bursts (in service and in the queue). Because it is difficult to directly characterize all these interacting traffic types, so that they are merged into a single measure in one step, the approximation is based on iterating two modules (described below) until a pre-assigned level of accuracy of the fixed-point solution is achieved. In addition to the fixed point numerical procedure associated with iterating the two modules, we have another numerical procedure that solves another set of fixed point equations within the second module. These two interacting iterative procedures introduce a certain level of complexity that, unfortunately, makes it difficult to provide a rigorous proof for the existence of a single fixed-point solution. Nevertheless, we have tested a wide range of system parameters and found that the approximation procedure in all cases converged to a single solution regardless of the initial conditions we chose.

The first module focuses on the traffic made of circuits with preemptive-priority over bursts and estimates its blocking probability. Notice that even this premium traffic does not enjoy strict priority over all other traffic streams as it cannot preempt non-premium circuits. Therefore, we cannot treat this traffic in isolation from the other traffic streams and therefore a fixed point solution is required. This first module obtains as input the blocking probability of the non-premium circuits. Having this blocking probability, it can estimate what we call *effective premium circuit* (EPC) traffic that can be considered in isolation, meaning that it is considered as if it has preemptive priority over bursts. This EPC traffic comprises the premium circuit traffic and the successful part of the non-premium circuit traffic. Accepting the assumption that the EPC traffic has preemptive priority over bursts, we notice that there is no difference from the point of view of the EPC traffic if a burst is being transmitted or being dumped on an input wavelength because in either case no new circuit can be transmitted on this wavelength during the time a burst is transmitted or dumped and in either case a new EPC arriving at another input wavelength can always be admitted. Including the burst transmission or dumping in the off time of the EPC traffic enables a single dimension analysis where the EPC traffic is considered in isolation in order to obtain the probability *p*_{j}
that there are *j* output wavelengths carrying EPC traffic in the system, *j*=0, 1,*K*, *K*+*N*. This gives us the blocking probability of the EPC traffic which is used to approximate the blocking probability of the premium circuit traffic. This turns out to be a very good approximation because the premium circuit traffic does not preempt the non-premium circuit traffic and beside the premium circuit traffic the EPC traffic includes only non-premium circuit traffic that has been admitted to the system, which can be considered in isolation from the burst traffic.

In the second module, we again consider a single dimension Markov chain representing the combined traffic of all types (circuits and bursts) where the on and the off periods are adjusted to include the results of the first module for the blocking probability of the premium circuit traffic and the effect of the dumped bursts. Using this single dimension system, we can derive expressions for the blocking probability for the bursts and any type *i* of non-premium circuits characterized by its *c*(*i*) value. Notice also that we found it necessary to treat the case of no circuit waiting *c*(0)=1 in a somewhat different way in module 2 when we derive the blocking probability of the non-premium circuits to improve accuracy.

We will now describe the two models in detail. In module 1, the effect of burst arrivals can be taken into account by modifying (increasing) the mean off period between two successive circuits [3], by setting the modified mean off period between two circuits be 1/*λ*′ as

where *λ*_{eff}
is the mean off period for the *effective total traffic* given by *λ*_{b}
+*λ*_{EPC}
and *λ*_{EPC}
is the mean off period for the EPC traffic given by (1-*B*_{nc}
)*λ*_{nc}
+*λ*_{pc}
(recall that *B*_{nc}
is the non-premium circuit blocking probability). Equation (4) can be rewritten as

The term 1/*λ*_{eff}
+1/*µ*_{b}
+1/*λ*′ in Eq. (4) is the mean off period given that the next arrival is a burst, which occurs with probability *λ*_{b}
/*λ*_{eff}
, while the term 1/*λ*_{eff}
is the mean off period given that the next arrival is a circuit (excluding those non-premium circuits which are blocked in the system with probability *B*_{nc}
), which occurs with probability *λ*_{c}
/*λ*_{eff}
. This gives rise to the following steady state equations. For 1≤*j*≤*K*,

and for *K*+1≤*j*≤*K*+*N*,

where *b*_{j}
=${\mathrm{\Sigma}}_{n=j-K}^{N}$
*c*(*n*). Together with the normalization equation

$\sum _{j=0}^{N}{p}_{j}=1,$ ,

the *p*_{j}
values can be obtained. The EPC offered load is given by

${O}_{\mathit{EPC}}=\sum _{j=0}^{K+N}\left(M-j\right)\left(\frac{\lambda \prime}{{\mu}_{c}}\right){P}_{j}$

and the EPC carried load is

${C}_{\mathit{EPC}}={p}_{\mathit{pc}}\sum _{j=1}^{K+N}j{p}_{j}.$ .

Thus, the EPC blocking probability (*B*_{EPC}
) and the premium circuit blocking probability (*B*_{pc}
) are estimated by

${B}_{\mathit{pc}}={B}_{\mathit{EPC}}=1-\frac{{C}_{\mathit{EPC}}}{{O}_{\mathit{EPC}}}.$ .

For the second module, as in [3], we argue that from the point of view of the switch, when a burst arrival is blocked and dumped, the input wavelength behaves as if it were inactive until the end of the burst has arrived at the switch. This can be considered equivalent to a situation whereby the blocked input wavelength having a longer off period with mean equal to 1/*µ*_{b}
+1/*λ*. This happens with probability *p*_{b}*B*_{b}
where *B*_{b}
is the burst blocking probability calculated later on. Let 1/*λ** be the modified mean off period; it is given by

Let *p*′
_{i}
be the probability that there are *i* bursts/circuits in progress/waiting. The modified mean on period, denoted 1/*µ**, is estimated by the following weighted average:

where *C*_{b}
is given by

${C}_{b}=\sum _{i=0}^{K-1}\left(M-i\right){p}_{b}\frac{{\lambda}^{*}}{{\mu}_{b}}{p}_{i}^{\prime}$

and

where

and

${C}_{\mathit{nc}}=\sum _{i=0}^{K-1}\left(M-i\right){p}_{\mathit{nc}}{p}_{c}\frac{{\lambda}^{*}}{{\mu}_{c}}{p}_{i}^{\prime}+\sum _{i=K}^{K+N-1}\left(M-i\right){b}_{i}{p}_{\mathit{nc}}{p}_{c}\frac{{\lambda}^{*}}{{\mu}_{c}}{p}_{i}^{\prime}.$ .

Note that when the ratio of *µ*_{b}
and *µ*_{c}
is large (say 100), the estimated value of non-premium circuit blocking probability, denoted as *B*_{nc}
, decreases drastically and becomes quite inaccurate when compared with the exact solution. Realizing that the non-premium circuit blocking probability *B*_{nc}
cannot be lower than the premium circuit blocking probability in the case where the circuits are given preemptive priority over bursts, denoted as *B*_{pc}
, we remedy this inaccuracy, by checking whether *B*_{nc}
, given by

${B}_{\mathit{nc}}=1-\frac{{C}_{\mathit{nc}}}{{O}_{\mathit{nc}}}$

where

${O}_{\mathit{nc}}=\sum _{i=0}^{K+N}\left(M-i\right){p}_{\mathit{nc}}{p}_{c}\frac{{\lambda}^{*}}{{\mu}_{c}}{p}_{i}^{\prime},$ ,

is lower than *B*_{pc}
. If it is found that *B*_{nc}
<*B*_{pc}
, we will bound the *B*_{nc}
value to *B*_{pc}
by setting *C*_{nc}
in Eq. (9) to *O*_{nc}*C*_{pc}
/*O*_{pc}
. Notice also that when circuit holding times are much longer than burst transmission times, namely, *µ*_{b}
/*µ*_{c}
is very large, given the fact that the circuits may queue, the system in most cases behaves very close to a system where all the circuits do have preemptive priority over bursts (See the Numerical Evaluation section for details). This means that in most cases when we need to use the preemptive priority bound, it is in fact a very tight bound.

This gives rise to a single dimension birth-death process described by the following steady state equations. For 1≤*i*≤*K*,

and for *K*+1≤*i*≤*K*+*N*,

and the normalization equation

$\sum _{i=0}^{K+N}{p}_{i}^{\prime}=1$

by which the *p*′
_{i}
values can be obtained.

The functional relation between *λ*′, *λ**, *µ** and *p*′
_{i}
expressed in Eqs. (5), (8), (9), (10) and (11) gives rise to a set of fixed-point equations. The fixed-point is computed by repeated substitutions.

We observe that when the buffer size is not zero, the non-premium circuit blocking probability is consistently higher than the exact value in our numerical examples. This is explained by the fact that in the real system some bursts should have been preempted by the premium circuits and would have given more buffer room to non-premium circuits. Therefore, we modify *B*_{nc}
for *N*>0 as follows:

where *p*_{pc}
is used to consider the effect mentioned above which is proportional to the probability that premium circuits appear and *B*_{pc}
is used to consider that the effect makes the nonpremium circuits behave more like the premium circuits. For the case of *N*=0, *B**
_{nc}
=*B*_{nc}
. Notice that the updated values of *C*_{nc}
and *C*_{c}
become *C**
_{nc}
=(1-*B**
_{nc}
)*O*_{nc}
and *C**
_{c}
=*C*_{pc}
+*C**
_{nc}
, respectively.

The blocking probability for circuits is given by

Then, we will calculate the blocking probability for bursts. The total load offered by bursts is given by

${O}_{b}=\sum _{i=0}^{K+N}\left(M-i\right){p}_{b}\frac{\lambda *}{{\mu}_{b}}{p}_{i}^{\prime}.$ .

Thus, the blocking probability for bursts is estimated by

${B}_{b}=1-\frac{{C}_{b}}{{O}_{b}}.$ .

The carried load of non-premium circuits in the queue is

${C}_{\mathit{qnc}}=\left(1-{p}_{\mathit{pc}}\right)\sum _{i=K}^{K+N-1}\left(M-i\right){b}_{j}{p}_{c}\frac{\lambda *}{{\mu}_{c}}{p}_{i}^{\prime}$

and the carried load of premium circuits in the queue is

${C}_{q\phantom{\rule{.2em}{0ex}}\mathit{pc}}={p}_{\mathit{pc}}\sum _{i=K}^{K+N-1}\left(M-i\right){b}_{j}\left(\frac{\lambda \prime}{{\mu}_{c}}\right){p}_{i}.$

Therefore, the carried load of circuits in the queue is

The proportion of circuits that are delayed, denoted as *D*_{p}
, is

${D}_{p}=\frac{{C}_{\mathit{qc}}}{{C}_{c}^{*}}$

The average number of circuits, denoted as *N*_{c}
, in the system is given by

where *N*_{pc}
is the average number of premium circuits, i.e.

${N}_{\mathit{pc}}={p}_{\mathit{pc}}\sum _{i=1}^{K+N}i{p}_{i},$

and *N*_{nc}
is the average number of non-premium circuits, or

${N}_{\mathit{nc}}=\left(1-{p}_{\mathit{pc}}\right)(\sum _{i=1}^{K+N}i{p}_{i}^{\prime}-{C}_{b})$

where the term ${\mathrm{\Sigma}}_{i=1}^{K+N}$
*ip*′
_{i}
is the average number of burst/circuit customers in the system and the term *C*_{b}
is the average number of bursts in the system.

By Little’s formula, the mean circuit delay is given by

${D}_{c}=\frac{{N}_{c}}{{C}_{c}^{*}{\mu}_{c}}.$ .

Notice that when the ratio *µ*_{b}
/*µ*_{c}
is large (say larger than 10) and Burst Proportion,

$\frac{\frac{{\lambda}_{b}}{{\mu}_{b}}}{\frac{{\lambda}_{b}}{{\mu}_{b}}+\frac{{\lambda}_{c}}{{\mu}_{c}}},$ ,

is small (say smaller than 0.7), the premium and non-premium circuit blocking probabilities become very close to each other and the estimated *D*_{C}
also becomes not as accurate as in other situations. We remedy the situation by treating all circuits as the premium circuits and obtain *D*_{C}
for that situation as follows:

${D}_{c}=\frac{{N}_{\mathit{pc}}}{{C}_{\mathit{pc}}{\mu}_{c}}$

The mean circuit queueing delay is

The mean queueing delay for a delayed circuit, denoted as *Q*_{qc}
, is given by

## 5. Numerical evaluation

Here we use the exact solution obtained by Eqs. (1), (2) and (3) to quantify the accuracy of our approximations. Results are presented here for the blocking probability and delay versus what we call the *normalized combined traffic intensity*, defined by

$\frac{M}{K}\left(\frac{{\lambda}_{b}}{{\mu}_{b}}+\frac{{\lambda}_{c}}{{\mu}_{c}}\right).$ .

Recall that the Burst Proportion (*BP*) is defined as

$\mathit{BP}=\frac{\frac{{\lambda}_{b}}{{\mu}_{b}}}{\frac{{\lambda}_{b}}{{\mu}_{b}}+\frac{{\lambda}_{c}}{{\mu}_{c}}}$

and the Premium Circuit Proportion (*PCP*) is defined as *PCP*=*p*_{pc}
. In all our examples, we consider 1/*µ*_{b}
=1 sec and for *N*>0 the *c*(*n*) values are evenly distributed, that is *c*(*n*)=1/*N*, for 0<*n*≤*N* and *c*(*n*)=0 otherwise.

In all scenarios studied regardless of the values of *M*, *N*, *K*, the traffic intensity, the ratio of *µ*_{b}
and *µ*_{c}
, *BP* and *PCP*, our numerical results show that the approximations in general agree well with the exact solutions as demonstrated, for example, in Figs. 3 to 10. The agreement demonstrated in Figs. 3 to 10 have also been observed in many other cases we considered; however, for brevity, we do not present them here.

Figure 3 shows the blocking probability versus the normalized traffic intensity under no buffer situation for *PCP*=0.3,0.7, *BP*=0.3,0.7 and *µ*_{b}
/*µ*_{c}
=10, respectively. It can be observed that under no buffer situation the blocking probabilities for bursts and non-premium circuits are indistinguishable.

Figure 4 shows the blocking probability versus the normalized traffic intensity varying *PCP* for *BP*=0.1 and *µ*_{b}
/*µ*_{c}
=10. It can be observed that the blocking probabilities for premium and non-premium circuits are indistinguishable. This suggests that for small *BP* using either premium or non-premium option for circuits does not make much difference.

Figure 5 shows the blocking probability versus the normalized traffic intensity varying *PCP* for *BP*=0.5 and *µ*_{b}
/*µ*_{c}
=10. It can be observed that the blocking probability for non-premium circuits decreases with *PCP* and the blocking probabilities for premium and non-premium circuits become indistinguishable when *PCP* is large.

Figure 6 shows the blocking probability versus the normalized traffic intensity varying *PCP* for *BP*=0.9 and *µ*_{b}
/*µ*_{c}
=10. It can be observed that the blocking probability for non-premium circuits decreases with *PCP*

Figure 7 shows the blocking probability versus the normalized traffic intensity varying *N* for *BP*=0.5, *PCP*=0.5 and *µ*_{b}
/*µ*_{c}
=10. It can be observed that both blocking probabilities for premium and non-premium circuits decrease with *N* without affecting much the burst blocking probability and delay for circuits. It implies that circuit queued reservation is an effective mean to reduce circuit blocking at almost no expense of burst blocking and delay for circuits.

Figures 8 to 10 show the blocking probability versus the normalized traffic intensity varying *BP* and *PCP* for *µ*_{b}
/*µ*_{c}
=100. It can be observed that if *BP* is not too large, the blocking probabilities for premium and non-premium circuits are indistinguishable. This suggests that for large value of *µ*_{b}
/*µ*_{c}
1 (say 100) and *BP* not too large (say less than 0.9) using either premium or non-premium option for circuits does not make much difference.

The number of states of the underlying Markov process is in the order of *MK*(*M*-*K*). In Fig. 11, we consider a more realistic case with *M*=100, *K*=50. For such a case, where we have hundreds of thousand of states, neither exact solution is possible, nor simulation results can give accurate results. However, our approximations can produce accurate results in less than a second. Figure 11 demonstrates how we can vary *N* to trade off between the burst blocking *B*_{b}
and circuit blocking (*B*_{pc}
or *B*_{nc}
). As demonstrated in Fig. 11, with slight increase of *N* from 0 to 5, both premium and non-premium circuit blocking probabilities reduce dramatically while the burst blocking probability increases slightly with small additional queuing delay for the circuits. This is in line with our intention in this paper to provide advantage to circuits over bursts by allowing circuits to wait for available wavelengths.

## 6. Network example

In this section, we illustrate how to apply our switch approximate model to a network environment and obtain the end-to-end blocking. We consider a dual ring network with 5 optical switches, each connected to an edge router. For each edge router, there are *M*=10 buffers for traffic with different destinations and with different QoS requirements and there are *K*=5 output wavelengths. In each direction, each link has *K*=5 wavelengths. We assume each edge router has the same external traffic, denoted as
$\overline{{\rho}_{x}}$
, destinating to every other optical switch. Figure 12 shows such a network. Note that
$\overline{{\rho}_{b}}$
,
$\overline{{\rho}_{\mathit{nc}}}$
, and
$\overline{{\rho}_{\mathit{pc}}}$
represent the burst, non-premium circuit and premium circuit traffic, respectively. Using the approximate model derived in the previous section, we can obtain the carried traffic of these three traffic types on the output link of the edge router and this carried traffic is in turn the input traffic of the optical switch connected to the edge router. Let
$\overline{{\rho}_{b}^{\prime}}$
,
$\overline{{\rho}_{\mathit{nc}}^{\prime}}$
, and
$\overline{{\rho}_{\mathit{pc}}^{\prime}}$
represent the burst, non-premium circuit and premium circuit traffic to each optical switch, respectively.

We assume that the source optical switch will choose the direction with the shortest path to the destination optical switch. We also assume that traffic is transmitted at each wavelength according to a finite on-off process, as mentioned before, and burst and circuit transmission times are independent and exponentially distributed.

Let us pick up an arbitrary link in the ring and consider all the routes traversing that link. Let *ρ*_{b}
, *ρ*_{nc}
and *ρ*_{pc}
denote the total burst, non-premium circuit and premium circuit load offered to each link, respectively. By symmetry, the blocking probabilities on each link are the same as given in the previous section. Consider the burst traffic. Summing the total carried burst load on a link and noting that it must equal (1-*B*_{b}
)*ρ*_{b}
, we arrive at the expression

or

For circuit traffic, we would instead have

where *x*=*n* or =*p*. Note that in circuit switching the carried load is not reduced at each successive link of a route. By assuming link blocking events occur independently from link-to-link, it can be easily shown that the average end-to-end burst blocking probability is given by

Similarly, we can obtain the average end-to-end circuit blocking probability

where *x*=*n* or =*p*.

The functional relation between *ρ*_{b}
, *ρ*_{nc}
, *ρ*_{pc}
, *B*_{b}
, *B*_{nc}
and *B*_{pc}
expressed in Eqs. (13) and (14), together with the relevant equations in the previous section, gives rise to a set of fixedpoint equations. As before, the fixed-point can be computed by repeated substitutions.

Figures 13 to 15 show the blocking probability versus the normalized traffic intensity for the network example varying *BP* and *PCP*. It can be observed that if *BP* is small (say 0.1) the blocking probabilities for premium and non-premium circuits are indistinguishable.

## 7. Conclusion

We have considered an optical hybrid switch implementation where circuits may have priority over bursts either by queueing circuits, or by providing circuits preemptive priority over bursts, or both. We have derived new approximations for the blocking probabilities of bursts and circuits, the proportion of circuits that are delayed and the mean delay of the circuits that are delay. The approximations are computationally scalable and accurate. These attributes enable the use of these approximations as part of a dimensioning tool used for practical scenarios involving hundreds or even thousands of wavelengths where neither exact solution is possible, nor simulation results can give accurate results.

## Acknowledgements

This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. 9041037] and by the Australian Research Council.

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