Geología de minas

In the LambdaPON network, connection requests and terminations arrive at random. For each request arriving, the NRM needs to resolve two issues

(i) the path or route to be allocated to the call and

(ii) the wavelength channel to be assigned.

These two problems are tackled in the remainder of this chapter.

As previously mentioned, each randomly generated connection request must be assigned a specific path or route through the network from the source to destination terminals and a wavelength pair. The wavelength pair assigned must be such that no

other connection on the same path, or on any path that shares an edge with the assigned path, is assigned the same wavelength.

A sim ilar path and routing problem exists in traditional circuit-sw itched telephone networks in which connections are established by selecting an available unused circuit through every link along the chosen route, from source to destination. The

selection of unused circuits is currently achieved by the use of switches [82]. If a

circuit does not exist the call is blocked, in which event, network blocking ^ is said to

have occurred.

In the LambdaPON, we must not only satisfy the constraints encountered by circuit- switched telephone networks, but also satisfy an additional constraint. Namely the same wavelength (analogous to a circuit in the circuit-switched telephone network scenario) must be assigned on every link on the selected route in order to establish the end-to-end connection between source and destination terminals. For the remainder

of the thesis this additional constraint shall be referred to as the wavelength continuity

constraint (as defined in [97]).

A

wavelength wavelength / o

wavelength Xy

B B

Figure 6.4 W avelength Continuity and Contention In The LambdaPON

In w avelength routed netw orks, w hilst unused circuits betw een the source and destination nodes may exist on a required route, the call might still be blocked so as to avoid violating the w avelength continuity constraint. In this case the resulting

blocking is a special case referred to as either wavelength contention or colour

clashing. The effect of colour clashing and the requirement for wavelength continuity 1 T he term "blocking" and "congestion" are equivalent; they refer to the condition in w hich a connection cannot be com pleted due to either the unavailability o f a free path or circuit caused by the need to share limited resources (fibre and switch) am ongst a large number o f term inals.

are shown in Figure 6.4. Consider three separate simultaneous sessions, AA', BB' and

CC allocated wavelength pairs Xj, ^2 and Xg respectively. The different wavelength

pairs have been represented by different colours. Note how the routes chosen and wavelength pairs allocated are such that they are colour-continuous and colour-

disjoint. If a new session DD' were requested, free circuits between D and D' do exist

such as wavelength pair Àg between node 2 and 5 and between node 5 and 6,

however no one continuous wavelength pair can be allocated without wavelength

contention. To avoid colour clashing either a new wavelength is required or a

wavelength converter at node 5 to convert Xg to X^. Notice, additionally, how in the

broadcast and select part of the LambdaPON network (namely within the PONs), wavelength reuse is not achievable and a different wavelength is required for each session.

Note that if dynamic wavelength translators / converters were incorporated into each of the nodes, such that a wavelength on an incoming link to a node could be translated to a different wavelength on the outgoing link, the wavelength continuity problem would no longer exist and the connection problem would be equivalent to that faced in current circuit-switched telephone networks. Unfortunately, the development of wavelength converters is still not mature technology. Also their use would add substantially to both the cost and the complexity of the network. The trade-off between increased complexity and cost, versus lower network blocking is studied in Section 8.10.

Since each fibre is capable of carrying several wavelengths, and different wavelengths do not interact with one another, we can imagine each wavelength as being one copy

of the network operating in parallel to each other. Hence a ring network with W

wavelengths available on each link can be physically represented as shown in Figure

6.5a and logically represented as W separate optical fibre rings as shown in Figure

6.5b.

Hence, the problem of wavelength assignment involves finding an end-to-end path and continuous wavelength pair from the source to the destination user. This can be viewed as the same problem as finding an edge disjoint route on one of the parallel network copies (figure 6.5b). Finding a route for a call implies specifying both the wavelength channels to be used and the path to be taken (i.e. the sub-nodes to traverse). Each session must have a path dedicated to it from the source to the destination node. Since one wavelength can carry only one signal on any link in any one direction, finding a disjoint path for a new session can be hard.

$

a) Physical Topology b) Logical Topology

Figure 6.5 Physical and Logical Representation O f The M ulti-W avelength LambdaPON Network

6.2.2 Static Versus Dynamic Routing And W avelength Assignm ent

The problem of finding both the wavelength channel to be used and the path to be taken can be resolved using any one of three different methods;

(a) Static path and wavelength assignment techniques

(b) Quasi-Static route and dynamic wavelength assignment

(c) Dynamic route and wavelength assignment

There are two distinct classes of static routing problems which can be encountered. (i) The first is the more common in which all possible paths between every source

and destination terminal is determined in advance and wavelengths preassigned to the routes during the network planning and design stages. An arriving request between source and destination is assigned one of the predetermined routes and, in doing so, this normally dictates the wavelength(s) to be used. This scheme is best im plem ented in an environm ent in which the traffic flow between term inals is known, constant and the connection times are long (of the duration days, months

or even years). This scenario typically arises in the long haul (trunk) network

where it is used to route traffic between Digital Main Switching Centres [981. However, purely static routing and static wavelength assignments algorithms are relatively inefficient in their distribution of network resources since they unable to take advantage of load variations which arise from business/residence, time zones and seasonal variations [991; nor are they flexible towards node or link failures. On the other hand, the control and management of systems based on static routing and assignment algorithms are generally greatly simplified relative to the next two

schemes presented. At present the disadvantages of static schemes seem to outweigh the advantages obtained by simplified control and consequently static routing, and channel assignment techniques are now rarely used.

(ii) The second form of static routing presupposes a prior knowledge of all connection requests which are likely to occur in the future; allowing the routes to be predetermined and established. Naturally, in a real network situation this is highly unlikely since calls are initiated and terminated at random. However, this option needs consideration because in the advent of ultra fast tuneable filters and tuneable lasers (of the order nanoseconds^), it is conceivable that calls in progress could be rerouted and wavelengths reassigned without the degradation of the connection or indeed even the customers' knowledge. The re assignment of wavelengths to calls can result in a call being serviced instead of being blocked [100]. This re allocation scheme can be viewed as a static routing problem since the network appears to be in a temporary frozen state in which all the calls can be reassigned new paths and wavelengths in such a manner as to maximise the networks resources (wavelengths and links in the case of the LambdaPON network). However, the advantages of applying this technique needs to be given careful consideration and the trade-offs between increased network complexity and processing versus increased traffic throughput quantified. The practicality and cost of this will be discussed further in Chapter 9.

Quasi Static routing and dynamic channel (wavelength) allocation is presently used to route most of the national and international traffic between the main switching exchanges. Here a limited degree of flexibility is available, and rearrangement of the network's routes between nodes can take place, albeit very slowly relative to the transmission speeds (over hours or even days). Quasi static routing normally presupposes the existence of spare capacity in the network which can be used to route information around a fault or congested route [49].

Dynamic routing schemes tend to be more flexible, make better use of network resources, but are more complex to operate. The purpose of dynamic routing schemes is to adjust routing patterns within a network in accordance with varying and uncertain offered traffic, to make better use of spare available capacity in the network resulting from dimensioning upgrades or forecasting errors, and to provide extra flexibility and robustness to respond to failures or overloads. The LambdaPON network makes use of a dynamic routing scheme which decides whether a requested session should be accepted, and, if so, how the call should be routed and which

^ There is normally a tradeoff between tuning speed and tuning range. At present both are achievable but unfortunately not in the same device.

wavelengths should be allocated. In a fully connected network the natural first choice route for a call is via the corresponding direct link. The im portant question for dynamic routing schemes then becomes what should be done when a call is blocked on its first choice route; should the call be accepted by the next longer possible routes and if several of these secondary routes exist which one should be chosen. In addition, if several wavelengths are available on a selected route, which wavelength should be allocated and does it make any difference to the network throughput. W hy chose one wavelength in preference to another?

6.2.3 The Com plexity In C alculating The B locking C haracteristics O f The

LambdaPON Network.

NETWORK

CONTROL B'

D'

Figure 6.6 Call Localisation and wavelength reuse in the LambdaPON

The wavelength blocking probability of the LambdaPON is dependent on the degree of wavelength reuse achievable. However, the degree of wavelength reuse is strongly dependent on the routing and wavelength allocation algorithms, the network topology and the traffic characteristics of the users on the network. The traffic characteristics

refers to the "traffic pattern" and the degree oi"call localisation". These two terms

originates and terminates on the same PON (such as AA', CC etc.). The larger the number of local calls, the greater the degree of wavelength reuse. In a ring topology

with N nodes, a call which originates on node 1 and terminates on node N/2 (i.e. the

source and destination terminals are on opposite sides of the ring such as DD'), occupies a wavelength pair over N/2 links and hence reduces the amount of wavelength reuse achievable.

Hence, the degree of wavelength reuse achievable (and hence the wavelength availability) is dependent on several inter-related variables:

(i) the topology of the network. At present we have discussed a simple bi­

directional ring topology, however a highly meshed architecture is preferred for

enhanced network throughput (as will become apparent in Chapter 8. The

higher the meshing the smaller the number of sub-nodes that need to be traversed in establishing a connection between a pair of users.

(ii) the "localisation" of the call.

(iii) the algorithm used to allocate the wavelengths. (iv) the routing algorithm used to find the route.

(v) the average duration of each connection which dictates how quickly each wavelength is released and therefore free to be reassigned to another connection request.

For small networks (less than 10 nodes with a few tens of wavelengths) with simple topologies, the blocking probability of the LambdaPON can be mathematically

computed for any one fixed known network trafGc state. As the network size

grows and the number of wavelength pairs in use increases the calculation becomes increasingly difficult. To fully understand the blocking characteristics of the LambdaPON, it is not sufficient to simply compute the blocking probability of any one fixed network state. To determine the blocking probability for random traffic patterns and to understand the effects of all altering all the variables listed above,

several, if not all (should this be possible), of the network states should be considered.

If we consider a simple bi-directional LambdaPON ring of N nodes and assume

(i) the availability of W wavelengths

(ii) the use of a shortest path routing algorithm

(iii) very low or zero blocking

(iv) the generation of only calls local to the PRN

The total number of possible network states for 1 wavelength channel is 2^. Hence,

is the number of states occuring due the establishment of all connections (W + l)

with a distance of 2 hops only and (W +1) is the number of states occuring due the

establishment of all connections with a distance of 3 hops only etc. Hence the total N number of possible network states for all possible calls is at least as large as (W+l) .

This problem can be categorised as a non-deterministic polynomial problem (NP

hard). A deterministic algorithm is one in which for any given state there is at most one valid "next" state. Conversely, a non-deterministic algorithm is one in which, for any given state, there may be more than one valid next state i.e. non-deterministic algorithms can do more than one thing at a time. A polynomial-time algorithm is a problem whose computational running time (that is the number of elementary bit operations it performs) is resolvable in polynomial time and the problem of

computing the problem is said to be NP complete [101]. In other words, a polynomial­

time algorithm is one which terminates in a number of steps bounded by Cn^ where C, k > 0 are constants and n is the size of the input to the algorithm (the number of nodes in the case of the LambdaPON).

Figure 6.7 shows a hypothetical algorithm for which the steps required for an input of size n is given by the left hand column Suppose that these are implemented in a computer which is capable of performing a single step in 0.00(XX)1 seconds. Figure 6.7 shows how the amount of time needed to solve instances of the problem increases with the size of the input. Notice how the two exponential time algorithms soon hit a point of exponential explosion making them utterly useless.

Time n = 10 Nodes n = 20 Nodes n = 40 Nodes n = 60 Nodes

n 0.00001 sec 0.0(XX)2 sec 0.(XX)04 sec 0.(XK)06 sec

n2 0.(XX)1 sec O.CKXM sec 0.(X)16 sec 0.(X)36 sec

n3 0.001 sec 0.(X)8 sec 0.064 sec 0.216 sec

n5 0.1 sec 3.2 sec 1.7 min 13 min

2n _0.001 sec ₁ sec 12.7 days 366 centuries

3" 0.0059 sec 58 min 3855 Centuries 1.3 X lOl^ millenia

Figure 6.7 Time Required to Solve An Algorithm Assuming A Single Step Can Be Computationally Performed in 0.000001 seconds.

Perhaps more illustrative is an example of a problem for which there is a fixed upper bound on the time in which it should be solved. Figure 6 .8 considers the same six algorithms but in this case the algorithms are being resolved using computers with

different processing capabilities. Note that upgrading the computer to one which has one thousand times the speed will allow the linear-time algorithm to solve problems

1000 times larger, whereas the 3° time algorithm can only solve problems whose size

increases by an additive constant.

Time With Processing Speed P with speed lOOP with speed lOOOP

n Ni 1 0 0 Nj 1 0 0 0 Nj n2 _N2 ION2 31.6 N2 N3 4.64 N3 1 0 N3 n5 _N4 2.5 N4 3.98 N4 2n _N 5 Ng + 6.64 N5 + 9.97 Ne N^ + 4.19 N^+ 6.29

Figure 6 . 8 Time Required to Solve An Algorithm Assuming Variable Computer

Processing Speeds

One of the best known problems which is N P hard is the "travelling salesman

problem" in which a travelling salesman has to visit n cities before returning to his starting point, whence his trip is complete. The algorithmic problem asks for the cheapest route, namely for a closed tour that passes through each of the cities and whose total distance travelled is minimal. Whilst resolvable for a small number of

cities the problem quickly scales up as n increase. In fact it has been shown that the

problem scales as »! [102].

Since the calculation of the overall average blocking probability of the LambdaPON

for all possible variation is NP hard , it is no longer possible to resolve the problem

mathematically for large networks (greater than 40 nodes) or for large numbers of wavelength pairs (of the order 30). It is in the authors opinion that the problem is best resolved by employing modelling techniques.

6.3 CONCLUSIONS

Contention for the Network Routing Manager can be detected using simple control protocols. In this case the ALOHA protocol was chosen because of its simplicity and prevalent use in modified forms. Contention for the Network Routing Manager, and hence the common control signalhng channel, is shown to be negligible assuming that each NRM is shared by around 10000 users who transmit control packets (containing about 200 bits of information) at a minimum bit rate of 2 Mbit/s and do not request more than 15 connections per minute. These figures are assumed typical for the successful design and implementation of the LambdaPON (Chapter 9).

The LambdaPON is a loss network i.e. a network in which the number of resources available is limited and hence at high traffic loading (relative to the number of resources available) a certain degree of connection requests will inevitably be blocked. In the LambdaPON these resources refer to the availability of a continuous and discrete wavelength pair on a chosen route for each connection request. The mathematical complexity in calculating the overall blocking of the LambdaPON is discussed and simulation techniques are suggested as an alternative means of obtaining an insight into the network blocking .

CHAPTER SEVEN