On Available Bandwidth in FDDI-Based Recongurable Networks. Sanjay Kamat, Gopal Agrawal, and Wei Zhao. Texas A&M University.

Transcription

1 On Avaiabe Bandwidth in FDDI-Based Recongurabe Networks Sanjay Kamat, Gopa Agrawa, and Wei Zhao Department of Computer Science Teas A&M University Coege Station, Teas Abstract The increasing use of high-speed networks in mission safety critica appications has ed to a growing concern for reiabiity issues in network design. The conventiona approaches to evauating network reiabiity in terms of their connectivity or mean time to faiure are often not sucient to characterize the network reiabiity. A better measure of system reiabiity shoud quantitativey reect trac carrying capacity of the network in the presence of fauts. In this paper, we present a probabiistic anaysis of the avaiabe bandwidth of FDDI-based recongurabe networks given the number of ink fauts. Three networks of varying degree of recongurabiity are considered: fuy recongurabe, partiay recongurabe, and non-recongurabe networks. The partiay recongurabe networks are found to be an eceent compromise between high reiabiity and the ease of impementation. A method of impementing the partiay recongurabe network using eisting FDDI technoogy is proposed. 1

2 Contents 1 Introduction Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Previous work : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : About this paper : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The network architectures Network mode : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Faut mode : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Reconguration capabiities of FBRNs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Three network architectures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Reconguration agorithm for an FBRN : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 3 Anaysis of avaiabe bandwidth Metrics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Computing A(; ) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Non-recongurabe network : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Fuy recongurabe network : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Partiay recongurabe network : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Theorem for computing AX(; ) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Numerica resuts : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Computing E[s j ] : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Non-recongurabe network : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Fuy recongurabe network : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Partiay recongurabe network : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Numerica resuts : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Taking wrap-around into account : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 4 Concusions 17 5 References 18 2

3 1 Introduction 1.1 Background The use of computer and communication networks in mission safety critica environments is becoming increasingy common. A major concern in the design of such networks is their vunerabiity to fauts or damages inicted by component faiures and/or enemy attacks. The objective of this paper is to address reiabiity issues in FDDI-based networks. Much of the work on network reiabiity has been based on connectivity. In this kind of studies, a network is considered operationa if it is connected, i.e., communication paths eist among a nodes. Measures such as the minimum or average number of fauts that wi disconnect the network are studied. However, in many appications connectivity aone is not sucient to assess the network operationa status. As pointed out by Wikov in [20], a reiabiity measure shoud quantitativey reect the trac-carrying capacity of a network in the presence of fauts. In this paper, we focus on the avaiabe bandwidth as a measure of the reiabiity of a network. We seected FDDI (Fiber Distributed Data Interface) networks for this study. FDDI is an ANSI standard for a 100 Mbits/sec ber optic token ring network [1, 2]. FDDI is suitabe for mission safety critica appications not ony because of its high bandwidth but aso due to its bounded access time and its dua oop architecture. Many new civi and miitary networks are being deveoped based on the skeeton of FDDI. Eampes incude the High-Speed Data Bus and the High-Speed Ring Bus (HSDB/HSRB) [17, 18, 19], the Survivabe Adaptabe Fiber Optic Embedded Network (SAFENET) [4, 9, 11], and Fiber Distributed Data Network (FDDN) [3]. In its basic conguration, an FDDI network consists of two counter-rotating oops. These two uni-directiona ber oops together constitute what is known as the FDDI trunk ring. This architecture provides certain faut toerant properties to FDDI networks. For eampe, most eisting FDDI networks use one oop as primary oop and the other as backup. In the event of a faut on the primary oop, the backup oop is used for transmission. However, this basic conguration may not be abe to toerate two or more fauts. That is, in the worst case two ink fauts wi disconnect the network. To overcome this probem, Raph, Ukrainsky, Scheack, and Weinberg [15] proposed an aternate path FDDI topoogy that uses two FDDI trunk rings. In their network, each station connects to an FDDI concentrator which subsequenty connects itsef to the two trunk rings. By utiizing the buit-in reconguration capabiity in a concentrator it was showed that the network can remain connected in the presence of more than two ink fauts. This soution has signicant practica vaue since it can be impemented using currenty avaiabe FDDI components. However, the avaiabe bandwidth characteristics of such networks during a fauty situation have not been anayzed. Generaizing the idea of aternate path FDDI topoogy, we consider an FDDI-based recongurabe network (FBRN) consisting of n nodes and r FDDI trunk rings. That is, each node is inserted into the r rings. Consequenty, each node has r inks incident on it on the eft and r inks incident on the right. A node has certain reconguration capabiity: a eft ink can be connected to one of m right inks. This is a generaization of the traditiona FDDI network and that proposed in [15]. In a traditiona FDDI network, r = 1 and m = 1 whie for the network considered in [15], r = 2 and m = 2. We are going to study the avaiabe bandwidth of an FBRN given that there are certain number of ink fauts. Ceary, the avaiabe bandwidth depends on not ony the number of FDDI trunk rings (r) but aso the reconguration capabiity of the network. We wi consider three FBRN architectures characterized by their degree of recongurabiity: 1

4 1. Non-Recongurabe Network (N-FBRN): The nodes in these networks have no reconguration capabiity. In other words, m = Fuy Recongurabe Network (F-FBRN): In these networks, a node can recongure the connection of any of its input inks to any of the output inks, i.e., m = r. 3. Partiay Recongurabe Network (P-FBRN): This architecture provides an intermediate degree of recongurabiity, i.e., 0 < m < r. In this paper we wi consider a particuar P-FBRN architecture with m = 2. These network architectures wi be further described in Section 2. In this paper, we wi anayze the avaiabe bandwidth characteristics of the above networks using the foowing measures: 1. The probabiity that the avaiabe bandwidth eceeds a specied vaue for a given number of ink fauts. 2. The epected vaue of avaiabe bandwidth, given the number of ink fauts. These metrics provide a generaization of the traditiona connectivity oriented metrics in the sense that zero avaiabe bandwidth impies that the network is disconnected. As epected, we nd that the fuy recongurabe network has the best avaiabe bandwidth characteristics. However, such a network is dicut to impement. We nd that partiay recongurabe networks, which can be reaized using the eisting commercia products, show a substantia improvement in reiabiity over the non-recongurabe networks. 1.2 Previous work An overview of the reiabiity provisions for the FDDI token ring can be found in [16]. The reiabiity of various network congurations supported by the FDDI standard is discussed in [13]. A comparative anaysis of various station attachment schemes based on the end-to-end user reiabiity and the mean time to faiure metrics is presented in [10]. An automatic faiure isoation and reconguration methodoogy for FDDI is described in [21]. Reiabiity anaysis for dua homing FDDI networks is presented in [12]. The use of two FDDI trunk rings to interconnect stations by concentrators was rst proposed in [15]. Our study etends their scheme to buid a mutipe ring FDDI-based network. In [6], Hies and Marow propose an FDDI concentrator tree topoogy with oopback which can toerate mutipe fauts. For a recent survey on various approaches for improving the reiabiity in FDDI networks, the reader is referred to [5]. 1.3 About this paper This paper is organized as foows. In Section 2, we dene the system mode and discuss three muti-ring architectures with dierent degrees of recongurabiity. Metrics for anayzing the avaiabe bandwidth in a fauty network are dened in Section 3. The performance resuts for these schemes with respect to these metrics are presented. We present our concusions in Section 4. 2

5 2 The network architectures 2.1 Network mode FDDI is a high-speed, ber-optic token ring network. The FDDI trunk ring actuay consists of two uni-directiona counter-rotating ber oops, each capabe of transmitting data at 100 megabits per second. Both the oops within an FDDI trunk ring can be used for data transmission at the same time providing a tota bandwidth of 200 megabits per second. However, most eisting FDDI networks use ony one oop for transmission reserving the other as backup. In the event of a faut on the primary oop, it is possibe to switch to the secondary oop. When a faut at a point of the trunk aects each of the oops, the network can perform a wrap-around, i.e, the primary and the secondary oops are wrapped to isoate the faut. The operation of wrapping resuts in a singe oop having a bandwidth of 100 megabits per second. Thus, the basic FDDI trunk ring provides some buit-in faut toerance capabiities. Note however, that more than one faut on each of the oops wi disconnect the network, dividing it into isoated segments. The need for providing greater bandwidth and increased faut toerance is the motivation for enhancing the basic FDDI topoogy. Both these goas can be achieved by using mutipe trunk rings to connect the individua nodes. Specicay, we use r FDDI trunk rings to interconnect n nodes. Each node has certain reconguration capabiity. Reconguration capabiity of a network comes into pay when the wrap around capabiity can not be used to recover from the fauts. The reconguration capabiity of the nodes makes it possibe to have a trunk segment beong to dierent trunk rings at dierent times. This faciitates isoation of fauty trunk segments in the network and utiization of faut-free segments to provide high avaiabe bandwidth. We ca such muti-ring networks FDDI based recongurabe networks (FBRNs). The detais of reconguration capabiities of FBRNs wi be discussed shorty. In an FBRN, each FDDI trunk ring is treated as one entity at east as far as reconguration is concerned. Reca that an FDDI trunk ring consists of two counter-rotating oops. In this paper, we wi assume that one oop within a trunk ring is used for transmission and the other is reserved as backup. However, our anaysis can be easiy etended for the case when both the oops are used simutaneousy. We denote the trac carrying capacity of a faut-free FDDI trunk ring as one unit. It is convenient to dene some terminoogy for FBRNs. To avoid repetitiousness, we wi use the term ring to mean an FDDI trunk ring. The section of a ring between two neighboring nodes on the network is caed a ink. With n nodes and r rings, there are n r distinct inks in the network. The coection of a the inks between two neighboring nodes wi be caed a bunch. Hence, there are n bunches in the network with each bunch consisting of r inks. 2.2 Faut mode Reiabiity issues in ring networks arise from ikey fauts in individua nodes and the inks connecting the nodes. The objective in designing a reiabe ring network is to maintain the connectivity of nodes in spite of such fauts when they occur. Node fauts in a ring can be handed by removing the fauty node from the ring. FDDI provides an optica bypass mechanism at a node which can be activated to remove that node from the ring. If a node performs some critica functions, then these functions must be dupicated esewhere on the ring so that a faut at that node does not disrupt the network service. Hence, in this study, we disregard node faiures and concentrate on ink fauts. Note that a ink was dened as a segment of an FDDI trunk ring. Hence, it consists of a pair of ber segments beonging to the two counter-rotating oops which constitute the ring. There are two 3

6 kinds of ink fauts: one in which ony one oop is aected and the other where both the oops are aected. The rst kind of ink faut may be handed within the FDDI trunk ring by hopping to the other oop if the other oop is faut-free. An FDDI trunk ring can recover from the second kind of ink faut by wrapping the two oops together on both the sides of the point of faiure. Note that this does not work if there are more than one such ink fauts on a ring. In this case, the reconguration capabiity of the nodes is used to recover from such fauts. In this paper, we concentrate on the second kind of ink fauts. Due to the spatia proimity of the two ber segments constituting a ink, it is ikey that a ink faut may aect both the segments. This is particuary true in a miitary environment where the fauts are a consequence of enemy attacks. We wi assume that a inks have the same probabiity of faiure. We consider a ink to be either in active state or in faied state. A ink in the faied state cannot carry trac. A ink is considered active if it is faut-free. We dene the state of the system as the combination of the states of a the inks in the network. Since there are n r inks, there are in a 2 possibe system states. Further, the number of system states in which there are eacty ink fauts is Reconguration capabiities of FBRNs We assume that initiay, a rings are functiona with each ring contributing one unit of bandwidth. Hence, the network can provide a maimum avaiabe bandwidth of r units when a inks are functiona. The avaiabe bandwidth reduces when some inks fai. The impact of such faiures on the avaiabe bandwidth depends on the abiity of the network to dynamicay recongure itsef in the event of ink fauts. We wi now consider three network architectures with varying degree of recon- gurabiity Three network architectures Two etreme cases of recongurabiity are described rst. Non-Recongurabe Network (N-FBRN). In this case, each ink between two neighboring nodes permanenty beongs to a distinct ring. If a ring has a fauty ink, it stops functioning. Fuy Recongurabe Network (F-FBRN). In this case, there is a switch at each node which makes it possibe to route messages from one trunk ring to another. As a resut, a ink may beong to dierent rings at dierent times depending on the conguration of the switches. The network management entity at each node tries to use as many active inks of a bunch as possibe. The dierence between two networks is iustrated in Figure 1. In this gure ring 3 and ring 4 have two fauty inks as shown. For a non-recongurabe network depicted in Figure 1(a), the avaiabe bandwidth is reduced from origina 4 units to 2 units, since ring 1 and ring 2 are the ony active rings. Further, note that neither ring 3 nor ring 4 can wrap around its two oops as both the rings have two fauts each. However, with fu reconguration capabiity at the nodes, nodes A and C are 1 p q denotes the number of combinations of p objects taken q at a time. 4

7 A A B D B D C (a) C (b) Figure 1: Eampe 1 A A B D B D C (a) C (b) Figure 2: Eampe 2 5

8 can be recongured to merge the segments of rings 3 and 4 as shown in Figure 1(b). Thus, for the same faut pattern, the avaiabe bandwidth is 3 units in an F-FBRN. It is evident that the addition of recongurabiity woud improve the avaiabe bandwidth of the network in the presence of fauts. However, a node in a fuy recongurabe network is compe, particuary when the number of rings is arge. However, in Appendi we wi show how a fuy recongurabe network of 2 trunk rings can be impemented by using eisting FDDI products. Using such smaer recongurabe networks as buiding bocks we can consider a third scheme which provides partia recongurabiity. Partiay Recongurabe Network (P-FBRN). This architecture is a compromise between the high eibiity of the F-FBRN architecture and the ow impementation costs of the N-FBRN architecture. In these networks, the nodes are connected using an even number of rings. The r rings are organized as r distinct ring-pairs. 2 Reconguration is possibe ony between rings within a ring-pair but not across ring-pairs. In other words, the n nodes are interconnected using r F-FBRNs each with two rings. 2 For the faut pattern of Eampe 1 shown in Figure 1, the avaiabe bandwidth of a partiay recongurabe network is the same as that provided by fu recongurabiity. However, Figure 2 iustrates an eampe where fu recongurabiity eads to higher avaiabe bandwidth than partia recongurabiity. In this eampe, a the four rings have ink fauts. With partia recongurabiity, segments of rings 1 and 2 or 3 and 4 may be merged by using the reconguration capabiity of the nodes. Figure 2(a) shows how rings 1 and 2 are merged to savage one unit of bandwidth. Note however that the specic ocation of ink fauts between nodes A and B precudes the merging of rings 3 and 4. Thus, with the partia recongurabiity, ony one unit of bandwidth is avaiabe due to the presence of fauts. However, with fu reconguration capabiity at the nodes, iustrated in Figure 2(b), two units of bandwidth are avaiabe by merging segments of rings 1 and 2 into one ring and those of rings 2 and 3 into another. Nevertheess, the partia reconguration network has the advantage that it can be impemented using eisting FDDI technoogy as discussed in the Appendi. We wi aso see ater that in most cases, the partiay recongurabe network has a much better avaiabe bandwidth characteristics than the non-recongurabe network, whie not being too far behind the fuy recongurabe network Reconguration agorithm for an FBRN Given an FDDI-based recongurabe network (FBRN), once the ink fauts are detected, a reconguration agorithm shoud be eecuted to decide how to recongure the system. The current FDDI standard has specied a faut detection process which is sti appicabe to our FBRNs. Here, we discuss a reconguration agorithm. The N-FBRNs have no provision for reconguration. P-FBRNs are buit using mutipe F-FBRNs with two rings. Hence, we ony need to consider the reconguration agorithm for F-FBRNs. The reconguration agorithm is eecuted at each node after the faut detection process has been eecuted. Hence, a the nodes have a knowedge of the state of the inks incident on them (i.e., fauty or active). Let Left N [i] and Right N [i] denote the inks from ring i incident on node N on its eft and right respectivey. The pseudo code of the agorithm is as foows. 1. For (i = 1; i r; i++) mark Right N [i] unassigned. 6

9 2. for (i = 1; i r; i++) if (Left N [i] is active) then a. connect Left N [i] to Right N [j] for the smaest j such that Right N [j] is active and has not been assigned, and b. mark Right N [j] assigned. 3. wrap together both the oops of unassigned inks. We note that some of the systems (e.g., some eisting FDDI trunk rings) may not have wrap up capabiity. For this kind of system, Step 3 of the above agorithm shoud be skipped. We are now going to derive the avaiabe bandwidth in an F-FBRN using the above reconguration agorithm for both cases of with or without the wrap up step. We need a notation rst. Reca that an FBRN architecture consists of n bunches with r inks per bunch. Let h 1 ; h 2 ; : : :h n?1 ; h n denote the number of fauty inks in a the bunches arranged in a non-decreasing order, i.e., h i h i+1 for i = 1; : : :; n? 1. THEOREM 2.1 The avaiabe bandwidth in an F-FBRN using the above reconguration agorithm is 1. (r? h n ) units if the wrap up option is not used; 2. (r? h n?1 ) units if the wrap up option is used. Due to the space imitation, the proof of the theorem is not presented here. The interested reader is referred to [8] for it. However, the theorem can be intuitivey epained as foows. When the wrap up option is not used, obviousy, the avaiabe bandwidth is at most (r? h n ) units, where h n is the number of damages in the maimay damaged bunch. The above theorem assures that this bandwidth is achieved by the recongurabe agorithm. With the wrap up of oops within a damaged trunk, each ring with ony one ink faut can be restored in step 3 of the above agorithm. It can be shown that the number of such rings is (h n? h n?1 ). Hence the avaiabe bandwidth with wrap up is (r? h n ) + (h n? h n?1 ) units or (r? h n?1 ) units. 3 Anaysis of avaiabe bandwidth In this section, we derive probabiistic characteristics of avaiabe bandwidth for the three network architectures described earier. 3.1 Metrics It is usefu to introduce the foowing notations. = number of fauty (damaged) inks. k = units of bandwidth that are unavaiabe due to ink fauts. s = avaiabe bandwidth. That is, s = r? k: (1) 7

10 Note that for a set of random ink fauts, s is a discrete random variabe. The vaue of s for a network depends on the number of random ink fauts () and their distribution in the network. We wi use f sj () to denote the probabiity mass function of s conditioned on the vaue of. That is, f sj () = Prob(s = j ): (2) a X (n; r; ; k) = tota number of system states with ink fauts such that eacty k units of bandwidth are not avaiabe in a network X. b X (n; r; ; k) = tota number of system states with ink fauts such that at most k units of bandwidth are not avaiabe in a network X. b X (n; r; ; k) and a X (n; r; ; k) are reated as a X (n; r; ; k) = b X (n; r; ; k)? b X (n; r; ; k? 1): (3) The subscript X in the above notations denotes one of the three architectures described in Section 2. We wi use etters N, P and F to correspond to networks N-FBRN, P-FBRN, and F-FBRN respectivey. We consider two metrics for studying the avaiabe bandwidth characteristics of a network: 1. The conditiona probabiity that at east units of bandwidth are avaiabe, given that the number of fauty inks in the network is, denoted A(; ). Formay, A(; ) = rx s= f sj () (4) 2. The epected vaue of avaiabe bandwidth, given that inks in the network are fauty, denoted E[s j ]. From the denition of epectation, E[s j ] is obtained as E[s j ] = rx =1 f sj () (5) As before, a subscript X wi be used to indicate the use of these metrics in the contet of a specic network X. In anayzing the overa reiabiity of a network, an important consideration is whether a nodes can communicate in the presence of network fauts. Whie maintaining connectivity among a nodes is necessary for the nodes to be abe to communicate, it is not sucient to ensure an acceptabe eve of service. The eve of service depends on the trac carrying capacity of the network. The above metrics emphasize this aspect of reiabiity. We now present the anaysis of the three FBRNs with respect to the above metrics. For the purpose of simpicity, we wi start with the case when the wrap up is not used. Etension of our anaysis to the case when the wrap up is used wi be discussed ater. Aso note that the vaues of metrics for the no wrap up case provide ower bounds for the wrap up case. 3.2 Computing A(; ) We wi now derive the rst metric dened earier i.e., the conditiona probabiity that the avaiabe bandwidth is at east given random ink fauts, for the three networks dened in Section 2. We wi rst derive combinatoria quantities a(n; r; ; k) and b(n; r; ; k) for the three networks. Then we use them to estabish the desired resut. 8

11 3.2.1 Non-recongurabe network The foowing emma provides the reevant resut for the N-FBRN i.e., the non-recongurabe mutiring network. LEMMA 3.1 a N (n; r; ; k) is given by a N (n; r; ; k) = r k kx i=0 (?1) i nk? ni k i : (6) Proof: Reca that a oss of k bandwidth units in a non-recongurabe network is caused by eacty r k fauty rings. These k fauty rings can be seected in ways, which is the rst factor in the right k hand side of (6). We now show that the second factor in (6) corresponds to the number of ways in which out of k r inks from k rings can be fauty, such that each of the k rings has at east one fauty ink. Given a particuar set of k rings abeed 1 k, et B i denote the set of system states in which at east one ink from ring i is fauty. Thus S k i=1 B i denotes the set of system states where a the ink fauts beong to the specic k rings. Since there are ink fauts, the size of this set is foowing discussion, we wi consider S k i=1 B i as the universa set. nk. In the Our interest is in nding the size of the set T k i=1 B i. Using the resuts from set theory, this can be obtained as foows. n( k\ i=1 B i ) = n([ k i=1 B i)? n(\ k i=1 B i) = = = = nk nk nk kx i=0? n([ k i=1 B i)? f? ( X 1ik k 1 (?1) i nk? ni n(b i )? nk? n X 1i<jk n(b i \ B j ) + X 1i<j<mk n(b i \ B j \ B m )? : : : g? k nk? 2n + k nk? 3n? : : : 2 3 k i : (7) This estabishes the desired resut Fuy recongurabe network It is dicut to obtain a cosed form soution for this network for arbitrary vaues of n and r. However, we can obtain a recurrence reation for b(n; r; ; k) as stated in the foowing emma. ) 9

12 LEMMA 3.2 b F (n; r; ; k) satises the foowing reation: b F (n; r; ; k) = 8 >< >: 0 > n k, P kj=0 b F (n? 1; r;? j; k) r j < k, k n k and n > 2: (8) For the base case of n = 2, we have b F (2; r; ; k) = kx j=ma(0;?k) r j r? j : (9) Proof: The rst boundary condition is easiy estabished using the pigeonhoe principe. If > n k, at east one of the n bunches must have more than k fauty inks. The second boundary condition foows from the fact that with < k we cannot have a bunch with more than k fauty inks. To derive the recurrence reation, consider the set of a system states with ink fauts which resut in a oss of at most k bandwidth units in an F-FBRN. This set can be partitioned into mutuay ecusive subsets based on the number of ink fauts from a particuar bunch. If this bunch has eacty j ink fauts r (which coud happen in ways), then the remaining? j fauts must be distributed in an identica j fuy recongurabe network with n? 1 nodes aowing for at most k units of oss in bandwidth. The base case for the recurrence is for a network with ony two nodes, i.e., n = 2. The epression for this case is obtained by noting that the system state is dened by the j fauty inks from one bunch and the corresponding? j fauty inks in the other. Note that neither j nor? j can eceed k. 2 The specia case of r = 2, is more amenabe to anaysis as seen from the foowing resut. LEMMA 3.3 a F (n; 2; ; k) for > 0 is obtained as foows a F (n; 2; ; k) = 8 >< >: 0 k = 0 or k > 2, n 2 k = 1, 2n? n 2 k = 2. (10) Proof: It is obvious that > 0 impies that the avaiabe bandwidth must reduce at east by one and hence a F (n; 2; ; 0) = 0. The case of k > 2 is trivia since there are ony two rings. To nd the number of system states which ead to oss of eacty one bandwidth unit, we observe that such states cannot have a bunch with both of its inks fauty. Since there are fauty inks, there must be n eacty bunches with one ink fauty in each of them. These bunches can be seected in ways and corresponding to each such seection, there are 2 ways of picking up the damaged inks in these 10

13 bunches. Hence, there are n 2 distinct system states corresponding to a F (n; 2; ; 1). Since, for non-zero, either one or two bandwidth units may be ost, and since ways in which out of the 2n inks may be fauty, a F (n; 2; ; 2) is obtained as 2 We wi nd this emma usefu in anayzing the partiay recongurabe network Partiay recongurabe network 2n is the tota number of The foowing resut estabishes a recurrence reation to compute a(n; r; ; k) for P-FBRN. 2n? a F (n; 2; ; 1). LEMMA 3.4 a P (n; r; ; k) satises the foowing recurrence reation X n a P (n; r; ; k) = a P (n; r? 2; ; k) + a P (n; r? 2;? j; k? 1) 2 j j j=1 ( ) X 2n + a P (n; r? 2;? j; k? 2)? 2 n j : (11) j j j=2 Proof: Reca that the partiay recongurabe network of r rings is composed of r pairs of rings with 2 each pair being a fuy recongurabe network. However, there is no switching from a ring from one pair to that from another. Consider the set of a system states in which inks are fauty and eacty k units of bandwidth are ost in case of P-FBRN. This set can be partitioned into three casses based on the status of one specic pair: 1. No ink fauts in this pair of rings. 2. Some ink fauts in this pair with eacty one unit of bandwidth ost. 3. Some ink fauts in this pair with eacty both units of bandwidth ost. In the rst case, the probem reduces to seecting fauty inks in a simiar network with one ess pair of rings and eacty k units of oss in bandwidth. The second and the third case simiary reduce to smaer equivaent probems. In these cases, we rst consider j(j > 0) ink fauts in this pair and use the resuts of emma 3.3 for mutipicative factors. Summing over a possibe vaues of j gives the desired resut Theorem for computing A X (; ) Given that we have obtained either a(n; r; ; k) or b(n; r; ; k) for each of the networks, we now can estabish the foowing theorem that uses the vaues of either of these combinatoria quantities to compute A(; ). 11

14 THEOREM 3.1 A X (; ) for any network X is given by A X (; ) = = r? X k=0 r? X k=0 a X (n; r; ; k) b X (n; r; ; k)? b X (n; r; ; k? 1) : (12) Proof: Let U X (n; r; ; k) denote the probabiity that eacty k bandwidth units are unavaiabe for a particuar network X, given that inks have faied. Hence A X (; ), the conditiona probabiity that the avaiabe bandwidth for network X is at east can be obtained in terms of U X (n; r; ; k) as A X (; ) = r? X k=0 U X (n; r; ; k): (13) Since there are n r inks in the network, is the tota number of congurations in which distinct inks may be fauty. Hence, the conditiona probabiity that eacty k units of bandwidth are unavaiabe is or equivaenty U X (n; r; ; k) = a X(n; r; ; k) ; (14) U X (n; r; ; k) = b X(n; r; ; k)? b X (n; r; ; k? 1) (15) where a X (n; r; ; k) and b X (n; r; ; k) are dened in Section Thus, A X (; ) for the three networks discussed in Section 2 can be computed using (12) in conjunction with the resuts of the emmas of this section Numerica resuts The numerica resuts for A(; ) for some network parameters are potted in Figure 3. The network is assumed to have 20 nodes with 6 rings (i.e., n = 20 and r = 6). From this gure, it is evident that the partiay recongurabe network has a signicanty better reiabiity than the non-congurabe network if the avaiabiity of 3 or more units of bandwidth is critica for the network. As seen from Figure 3, a faiure of 10 inks (which constitutes approimatey 8% of the tota of 120 inks in the network) impies that the probabiity that at east two units of bandwidth are avaiabe is very ow for the non-recongurabe network. For the same conditions, the partiay recongurabe network has a much better performance. Simiar observations can be made from Figure 4. However, as seen from Figure 5, a more stringent requirement such as a high probabiity of having 4 out of 6 units avaiabe can be satisfactoriy met ony using a fuy recongurabe network. 12

15 1.00 A(2,) N-FBRN P-FBRN F-FBRN Figure 3: A(2, ) for dierent networks 0.80 A(3,) N-FBRN P-FBRN F-FBRN Figure 4: A(3, ) for dierent networks 13

16 A(4,) N-FBRN P-FBRN F-FBRN Figure 5: A(4, ) for dierent networks 3.3 Computing E[s j ] The epected vaue of avaiabe bandwidth s, when r random ink fauts occur, can be obtained as E[s j ] = = = = rx s=1 rx k=0 rx k=0 rx k=0 s f sj (s) (16) (r? k) U(n; r; ; k) (17) a(n; r; ; k) (r? k) (18) b(n; r; ; k)? b(n; r; ; k? 1) (r? k) (19) However, a dierent approach towards computing the mean avaiabe bandwidth eads to simper epressions. The foowing theorems provide the resuts for various network architectures of interest Non-recongurabe network THEOREM 3.2 For the non-recongurabe network, the epected vaue of the avaiabe bandwidth, 14

17 given that there have been random ink fauts, is given as foows: E N [s j ] = r? n : (20) Proof: Consider the r rings to be abeed from 1 to r. Let s i denote a discrete random variabe dened as ( 1 if ring i has no fauty inks s i = (21) 0 otherwise. Hence the avaiabe bandwidth s is obtained as s = rx i=1 Note that a s i 's are identicay distributed. Hence, E[s j ] = rx i=1 s i : (22) E[s i j ] = r E[s i j ]: (23) From the denition of s i, it foows that E[s i j ] is same as Prob(s i = 1 j ink fauts). This probabiity is easiy seen to be? n. Hence we obtain the resut (20) Fuy recongurabe network The genera case for this network is dicut to anayze. Numerica vaues of the mean avaiabe bandwidth can be obtained using (16). However, we have the foowing usefu resut for the specia case when r = 2. THEOREM 3.3 For the fuy recongurabe network with r = 2, the epected vaue of the avaiabe bandwidth, given that there have been random ink fauts, is E F [s j ] = 8 >< >: n 2n 2 > 0, 2 = 0 : 2 Equating the epression for the mean avaiabe bandwidth given by (20) with that obtained using (16) and the resuts of emma 3.1 gives us the foowing interesting combinatoria identity rx r?s? X n r r = s (?1) i? ns? ni s s=0 i=0 As yet, we do not know of any other proof for this identity. r? s i : (24) 15

18 N-FBRN P-FBRN F-FBRN E[s ] Figure 6: Mean Avaiabe Bandwidth Proof: The case of = 0 is trivia. For > 0, the resut foows by direct appication of (16) with the probabiities obtained using (14) and by emma Partiay recongurabe network THEOREM 3.4 For the partiay recongurabe network, the epected vaue of the avaiabe bandwidth, given that there have been random ink fauts, is E P [s j ] = r 2 P q=1 n q? 2n 2 q? q + r? 2n : (25) Proof: This resut can be proved using simiar reasoning as in Theorem 3.2 with s i being dened for the ith pair of rings (i.e, rings 2i? 1 and 2i) Numerica resuts The performance of the three networks based on the mean avaiabe bandwidth is shown in Figure 6. These gures are based on the same network parameters as before, i.e., n = 20, r = 6. The foowing observations can be made from Figure 6: 16

19 For a three network architectures, the mean avaiabe bandwidth decreases with increase in the number of fauty inks () as epected. For a sma vaue of, a three systems have comparabe performance. The abiity of a network to recongure itsef is tested ony when the number of ink fauts is sucienty high. On the other hand, a very arge vaue of means a heavy damage to the network and a the reconguration schemes tend to have a ow avaiabe bandwidth as is increased beyond a certain vaue. As epected, the mean avaiabe bandwidth for the fuy recongurabe system is the highest whie that for the non-recongurabe system is the owest. However, the partiay recongurabe system which is much easier to impement than the fuy recongurabe system has a very good performance. It is seen that partiay recongurabiity provides a signicant improvement over the non-recongurabe mutipe ring architecture. For eampe, when = 20, i.e., when 16:67% of the inks are fauty, the N-FBRN has an amost zero mean avaiabe bandwidth. On the other hand, for the same conditions, the mean avaiabe bandwidths of P-FBRN and F-FBRN are cose to 2 units and 3 units respectivey. 3.4 Taking wrap-around into account The anaysis presented so far did not consider the possibiity of wrapping around the two counterrotating oops of a trunk to recover from a singe point trunk faut. Etension of our anaysis to take this possibiity into account is possibe, athough the corresponding derivations become tedious. The genera approach is to consider the number of rings with singe faut separatey. The simpest case is that of deriving the mean avaiabe bandwidth for the non-recongurabe network i.e., E N [s j ]. To take wrap around in account, in Theorem 3.2, we change the denition of s i, the bandwidth contributed by ring i so that s i = 1 if ring i has 0 or 1 fauty inks. From straight forward combinatoria arguments, the probabiity of s i being 1 can then be obtained as Prob(s i = 1 j ink fauts) =? n Mutipying this probabiity by r gives the mean avaiabe bandwidth. + n? n? 1 : (26) For the genera case of fuy recongurabe networks, Theorem 2.1 indicates that consideration of wrap around impies that the mean avaiabe bandwidth is r? E[h n?1 j ], where h n?1 denotes the second argest number of inks damaged in a bunch. Reca that the sequence h 1 ; h 2 ; : : :h n denotes the number of inks damaged in the n bunches sorted in non-decreasing order. h n?1 is the second order statistic. An approach woud be to derive the distribution or the epectation of order statistics for these random variabes. For a more detaied discussion on these issues, see [8]. 4 Concusions In this paper we proposed the use of mutipe FDDI rings with recongurabe architecture. Our objective is to provide a signicant improvement over the standard FDDI networks in terms of connectivity and avaiabe bandwidth. We studied three FDDI based recongurabe networks varying in the degree of recongurabiity supported by the network nodes. We derived the probabiity of the avaiabe bandwidth eceeding a certain given vaue and the mean vaue of avaiabe bandwidth for 17

20 a given number of ink fauts. One of the important observations is that a partiay recongurabe network architecture provided a much better performance than a non-recongurabe network. The partiay recongurabe network is particuary attractive because it can be reaized using commerciay avaiabe FDDI products. The work done here can be etended in severa interesting ways. For eampe, dierent partia recongurabe networks other than the one discussed here can be investigated for their reaizabiity using currenty avaiabe FDDI products. Currenty we are working on the design and anaysis of a concentrator which can connect four FDDI trunk rings. Use of such a concentrator in a P-FBRN woud certainy resut in better performance than the one we considered in this paper. References [1] FDDI Token ring media access contro (MAC). ANSI Standard X3.139, [2] ANSI Standard X3T9.5, Fiber Distributed Data Interface (FDDI) - Token Ring Medium Access Contro (MAC) May, [3] M. D. Cohn, \A network architecture for advanced aircraft", Proc. IEEE Conf. on Loca Computer Networks, pp , Minneapois MN, Oct , [4] D. T. Green and D. T. Marow, \SAFENET { A LAN for navy mission critica systems", Proc. of 14th Conf. on Loca Computer Networks, [5] W. Hies, and D. Marow, \Approaches for survivabiity in FDDI networks", Proc. of 17th Conf. on Loca Computer Networks, [6] W. Hies, and D. Marow, \Eperimentation on the concentrator tree with oopback", Proc. of 18th Conf. on Loca Computer Networks, [7] M. J. Johnson, \Reiabiity mechanisms of the FDDI high bandwidth token ring protoco", Computer Networks ISDN Systems, Vo. II., No. 2, pp , [8] S. Kamat, \Performance issues in high-speed rea-time networks", PhD Thesis, under preparation. [9] R. J. Kochanski and J. L. Paige, \SAFENET { The standard and its appication", IEEE LCS, Vo. 2, No. 1., pp , Feb [10] D. Logothetis and K. Trivedi, \Reiabiity anaysis of various station attachment schemes in a FDDI token ring", [11] U.S. Department of Defense. \Survivabe Adaptabe Fiber Optic Embedded Network ", Sept MIL-STD-2204 [12] G. Nguyen, \Reiabiity anaysis for FDDI dua homing networks", Proc. of 18th Conf. on Loca Computer Networks, [13] K. B. Ochetree, \Using redundancy in FDDI networks," Proc. IEEE Conf. on Loca Computer Networks, pp , [14] Lt. J. L. Paige, \SAFENET - A navy approach to computer networking", Proc. IEEE Conf. on Loca Computer Networks, pp , Minneapois MN, Sept Oct. 3, [15] S. Raph, O. Ukrainsky, R. Scheack, and L. Weinberg, \Aternate path FDDI topoogy ", Proc. IEEE Conf. on Loca Computer Networks, pp , Sept [16] F. E. Ross, \An overview of FDDI: The Fiber Distributed Data Interface", IEEE Journa on Se. Areas in Comm., Vo. 7, Sept [17] SAE, Aerospace Systems Division, Committee AS-2, \Linear Token-passing Mutipe Data Bus", AS4074.1, Version 4.0, [18] SAE, Aerospace Systems Division, Committee AS-2, \High Speed Ring Bus (HSRB)", AS4074.2, Jan. 27, [19] R. W. Uhhorn, \The Fiber-Optic High-Speed Data Bus for a new generation of miitary aircraft", IEEE LCS, Vo. 2, No. 1, Feb [20] R. Wikov, \Anaysis and design of reiabe computer networks", IEEE Transactions on Communications, Vo. COM-20, pp , June

21 [21] Y. Y. Yang and R. Sankar, \Automatic faiure isoation and reconguration", IEEE Network, Vo. 7, No. 5, pp , Sept