Optimal Routing and Scheduling for Deterministic Delay Tolerant Networks

Transcription

1 Optimal Routing an Scheuling for Deterministic Delay Tolerant Networks Davi Hay Dipartimento i Elettronica olitecnico i Torino, Italy hay@tlc.polito.it aolo Giaccone Dipartimento i Elettronica olitecnico i Torino, Italy paolo.giaccone@polito.it Abstract Delay Tolerant Networks (DTNs), in which contacts between noes come an go over time, is a promising approach to moel communications in mobile a-hoc networks, where scenarios of network partitioning an noe isconnections are likely to happen. One of the most important challenges in such networks is how to route information an scheule transmissions, coping with the continuously changing network topology. In this paper, we focus on a eterministic an centralize DTN in which the contact times between noes are known in avance or can be preicte; this moel is applicable for various real-life scenarios. We provie a general framework for evising optimal routing algorithms to such networks uner ifferent objective functions an various real-life constraints (such as the available buffer an energy). The key insight is to moel the DTN as an equivalent time-inepenent graph; this allows the usage of well-known algorithms an techniques to achieve optimal results. These algorithms can be also use as approximation for less certain settings or as benchmarks to evaluate other routing algorithms. In aition, we extene our framework to eal with long-live DTNs in which contacts are perioic. Our algorithms are emonstrate by simulations, base irectly on real-life traces, showing capacity-elay traeoffs an the influence of the constraints an perioicity on the achievable throughput of the network. I. INTRODUCTION In Delay (or Disruption) Tolerant Networks (DTN), ento-en communications are not always possible on a irect routing path through the network, since the network connectivity is intermittent an at any instant the network may become partitione. Thus, routing is realize by multi-hop transmissions, exploiting spontaneous connections among the noes. The DTN moel is especially appealing in large mobile ahoc wireless networks (MANETs) that have no fixe infrastructure. In such networks, performance scalability is achieve by exploiting the mobility of the noes in orer to transmit ata in a store, carry an forwar manner [1]: ata is sent by a noe to a close-by noe, which stores it in its buffer; then, the intermeiate mobile noe carries the ata from one physical location to another an finally forwars it to its estination. This transmission paraigm is easily moele through a DTN, in which contacts between two noes appear in the DTN when the noes are physically close-by. This work was supporte by the Italian Ministry of University an Research (MIUR) through the RIN project CARTOON. The promising performance results an concise moeling have le to an extensive research on DTN uring the last few years (e.g, [2] [5]). At the core of this research line are routing an scheuling algorithms: at any given time, each noe shoul fin when an where to forwar the ata store in its buffer so that it reaches the estination in a timely manner. It is important to notice that the performance of the DTN are irectly ictate by the inter-contact times between the noes (which, in turn, are the result of the mobility pattern of the network). In particular, packet elays become comparable with these inter-contact times, implying that application running on DTNs shoul be tolerant to elays of orer of minutes/hours. rime examples in which DTN networks are use is when mobility comes for free : vehicular networks [6] (in which ata is carrie over cars an buses), pocket area networks [7] (in which ata is carrie by people carrying small evices like DAs), mixe groun/satellite networks an sensor networks. Note that the DTN moel captures also scenarios in which some noes are mobile an some noes are fixe (e.g, mobile evices with fixe gateways). This paper focuses on eterministic DTNs in which contact times are known (or can be accurately preicte) in avance. We note that this is not always the case, an we consier the problem of builing the DTN from a specific mobility pattern as orthogonal to this paper. However, in most of the contexts, the mobility, even if ranom or not perfectly known, is quite constraine in the space (like, following some specific routes, or having some spatial limitations like roas) an in the time (like, following some time scheule or some habit rule). Very recently, [8] has shown that also human mobility, consiere traitionally almost unpreictable, shows on the contrary simple an preictable time/space patterns, most of the time. In general, these spatial/temporal constraints are the key issues to consier when esigning practical DTN networks, proviing some level of performance guarantees (QoS) to the en users. Thus, investigating a eterministic DTN can serve as approximation an/or reference moel for less certain scenarios. A. Our contribution In this paper we aress the routing an scheuling problem in eterministic DTNs; namely, fining the sequence of noes on which ata shoul follow from the source to the estination,

2 an computing the contact times in which this ata shoul be transmitte. Due to the eterministic nature of our problem, we were able to evise a centralize optimization framework which computes polynomially the optimal performance of the network, in the spirit of evaluating the achievable level of QoS. When contact times are not known in avance or when centralize algorithms are not possible, our results can be consiere as a reference benchmark for performance comparison. Taking a ifferent perspective, given some QoS constraints, if our optimization framework is not able to fin a solution compatible with them, base on full contact time knowlege an complete scheuling control, it is not possible to evise any scheme to satisfy such constraints. In general, the core of our framework consists in constructing a time-inepenent graph expressing the chronological orer between contacts. Then, optimization problems are solve on this graph using classic algorithms from Graph Theory. It is important to note that previous works (e.g. [2] [5]) suggeste tailore-mae algorithms for each objective function, which works irectly on a particular time-epenent graph, a.k.a. DTN graph. Both the time an space requirements of our construction are linear in the number of contacts, thus o not introuce extra complexity on the classical algorithms. Our approach is preferable since it provies a general reuction to known problems an therefore can be easily applie to many objective functions an constraints as the nee arises. For example, we are able to moel easily the following performance costs: minimum elay (work-span), maximum banwith, minimum number of hops; in aition, we consier the following constraints: minimum banwith per flow, maximum buffer occupancy per noe, an maximum available energy per noe. Note that the last two constraints appear only for several scenarios in which DTN is usually consiere: energy is very important for DAs, cell-phones, but irrelevant for roa vehicles; on the other han, buffer sizes are not usually a constraint in wireless networks, but in some particular sensor evices it can be very small (like in imotes [9]), or in cooperative networks with selfish users, there might be some limitations on the size of the buffer evote to altruistic forwaring. Moreover, our constructions can easily moel the problem of multicasting an simultaneous source-estination flows as classical maximum multi-commoity flow an minimum Steiner tree problems. Thus, we can use the extensive research tackling these problems (an their many variants) in Graph Theory an apply it irectly on DTN. Finally, it is important to note that in this paper we eal only with atomic contacts, assuming that the contact urations are negligible with respect to the inter-contact times. This assumption is realistic in many scenarios, in which simultaneous contacts o not exist. B. revious work Routing problems in DTNs have been extensively investigate in recent years (see [5] for a reasonable set of references on these problems an other DTN-relate esign issues). In general, following the taxonomy in [10], routing algorithms can be classifie either as replication schemes, which sen many copies of the same ata packet across the network, or forwaring schemes, which sen only a single copy of each packet across the network. This paper eals with such routing algorithms an therefore in the sequel we will focus only on relevant results regaring forwaring schemes. It is important to notice that since in these schemes only a single copy is sent, their efficiency strongly relies on the knowlege of future contacts across the entire network. Seminal paper [2] has efine the routing problem in DTN an aresse many esign issues highlighting ifferent approaches base on several levels of knowlege within each noe. The original efinition of the DTN graph, base on the same assumptions as in our work, appears in [2]; as we have alreay observe, such construction is very useful to efine the routing problem, but, since it is base on a time-epenent graph, it requires the esign of a-hoc algorithms; inee, to compute the minimum elay path in the DTN graph, [2] proposes a moifie version of the Dijkstra algorithm. An alternative moeling of the routing problem was propose in [11], which efines a particular space-time graph on which the minimum elay path can be foun exploiting a classic shortest path algorithm. Specifically, the space-time graph hols a layer of the original DTN graph for each timeslot, such that two noes are connecte at some layer t if an only if there was a contact between them at time t; furthermore, copies of the same noe are connecte across layers. It is important to notice the size of the space-time graph (as well as the time it takes to run algorithms on it) epens on the overall time perio T in which the DTN is active. This results, taking the example in [11], in a weaklypolynomial all-pairs shortest-path algorithm with complexity O(n T ), where n is the number of physical noes. On the other han, our construction is linear with the number of contacts an in fact oes not introuce extra complexity to the classical (strongly-polynomial) algorithms. We note that a similar (weakly-polynomial) construction was consiere also in Graph Theory research in the context of network flows over time (a.k.a ynamic networks). Unlike the classical static network flow moel, this moel assumes flows requires preefine amount of time to traverse over an ege in the graph. As a consequence, the flow value over each ege can change over time in orer to maximize the total flow. The moel (an that construction) was first consiere by For an Fulkerson [12]; references [1], [14] provie goo surveys on the avances in this topic over the last ecaes. More recently, [] has consiere the scenario in which the mobility pattern of the noes is eterministic an perioic. As in [2], the authors compute the minimum elay (workspan) path using an extene version of Dijkstra s algorithm. Note that [] is more focuse on a novel hierarchical routing scheme, base on multilevel clustering, than on the routing algorithms. An interesting approach, extening the moel of perioic

3 mobility, has later been propose in [4], which consiers the minimum elay routing problem in the case of perioic but not eterministic mobility. Specifically, in this case each contact is characterize not only by a contact-time but also by a probability that the contact event will inee occur. The solution of the problem is obtaine using a construction that partially resembles ours. Inee, [4] starts from a probabilistic space-time graph an then buils a probabilistic space-space graph which is time-invariant; on this new graph, the optimal routing is evaluate as a solution of a Markov ecision process. Note that this construction has been propose for the minimum elay path only. Finally, [15] has consiere the same eterministic mobility of our paper an has propose some new algorithms to solve the routing problem (also in the multi-commoity flow scenario) taking into account similar constraints an performance costs as in this work. The algorithms in [15], which were erive from the algorithm in [2], run specifically on the DTN graph an their performance are not guarantee to be optimal. On the contrary, our optimization framework works on a timeinvariant graph, on which well-known algorithms are use to fin optimal solutions for the problem an all its variants. II. MODEL DEFINITIONS Let D = V, E be a elay-tolerant network multi-graph (a.k.a. DTN Graph) in which V enotes the participating noes an E enotes the time-contact eges: each ege e E has a label t(e) R + which specifies its contact time; in aition, let bw(e) R + {+ } enotes the banwith of the contact (namely, the number of bytes that can be transmitte between the noes exploiting the contact). We assume that transmissions o not overlap in time or, equivalently, that t : E R + is an injective function an that transmissions are instantaneous. In aition, we consier some resource constraints on the noes of the DTN graph: for each v V, let buf(v) R + {+ } enote the buffer size resiing at noe V an let en(v) R + {+ } be the total amount of energy (battery) available at the noe for transmissions. The goal of the DTN is to eliver a certain amount of bytes from one noe s V to another noe V. The bytes are elivere along (possibly many) paths = {p 1,... p m } between s an, which is the ecomposition of the flow on D to paths. Each path is enote by alternating sequence of noes an eges: p i = (s, e 0, v 0, e 1, v 1,..., e in 1, v in 1, e in, ). The time associate with the routing scheme is the maximum time associate with a link in one of its paths, namely t( ) = max{t(e j ) e j p i, p i }; this quantity is sometimes calle also the work-span of the routing scheme. It is important to notice that since each ege is associate with a specific time, a path p i ictates not only the routing flow of the ata but also the scheuling ecisions within each noe. We call the routing scheme of D an when it is clear from the context, with a slight abuse of notations, we also use to enote its corresponing flow. A feasible routing scheme of x bytes between s an must satisfy the following constraints: Chronological Orer: For each p an for each j {0, i n 1 }, t(e i ) t(e i+1 ). Flow Feasibility: The corresponing flow of is a legal flow on D, whose value is at least x. Buffer Constraints: Let occ(v, t, ) be the number of bytes store at noe v at time t uner routing scheme. For all v, t, occ(v, t, ) buf(v). Energy constraints: Let recv(v, ) an sent(v, ) be the number of bytes receive by noe v uner routing scheme an the number of bytes sent by noe v uner routing scheme. For every noe v, α recv(v, ) + β sent(v, ) en(v), where α an β are parameters of the problem instance. For simplicity, we normalize α an β such that α + β = 1. In aition, since is a flow, if v / {s, } then recv(v, ) = sent(v, ), thus the energy constraint can be rephrase as recv(v, ) en(v) (unless v = s or v = ). III. EVENT-DRIVEN GRAH CONSTRUCTION Our primary metho in orer to eal with problems on DTN graphs is to reuce them into time-inepenent graphs which capture the original constraints of the DTN. Then, the routing an scheuling problems on the DTN can be easily solve on these graphs, using well-known algorithms an techniques (such as augmenting path algorithms an linear programming). We first introuce the notation of an event in the DTN. In irecte DTN, a noe v V is associate with a sening event at time t if there is an outgoing contact from v at time t. Similarly, a noe u V is associate with a receiving event at time t if there is an incoming contact to u at time t. Thus, each contact in a irecte DTN will result in two events, one in its sening en an the other on the receiving en. In an unirecte DTN, each contact will result in four events, since each en can both receive an sen ata. The key concept of the construction of the time-inepenent graph is to represent each event in the system as a noe. Then, two such event-noes are connecte if either they are the sening an receiving event of the same contact (we call this type of connections inter-eges) or if both events occur on the same DTN noe an no event in that noe occurs between them (we call this type of connections intra-eges). Our construction is illustrate by a simple example in Fig. 1. We next given a formal efinition of constructing the graph for a irecte DTN; generalizing the construction for a unirecte DTN will follow. Given an irecte DTN graph D = V, E, let event-riven graph G(D) = V, E be constructe as follows: Noes: For each noe v V, let E v E be the set of eges touching noe v an efine the following set of noes V v = { v, t e E v, t = t(e)}. The set of noes in the event-riven graph is efine as V = v V V v. Furthermore, in the sequel, we say that a noe v, t belongs to a super-noe v if v, t V v.

4 D (t = 0, s bw = 10) (t = 5, bw = 4) u (t = 1, (t = 2, (t =, (t = 10, v w z bw = 5) bw = 2) bw = ) bw = 5) buf = 4 buf = buf = 5 buf = 2 buf = 7 buf = (t = 8, bw = 10) D (t, bw) u buf = bu v buf = bv G(D) u, t, 0 bw bu u, t, 1 v, t, 0 bw bv v, t, 1 bw = 10 s, 0 u, 0 v, 1 w, 2 z, bw = 5 bw = 2 bw = bw = bw = 5 bw = 2 bw = 7, 10 Fig. 2. Example of constructing event-riven graph G(D) from a elaytolerant network D with a single contact. G(D) u, 1 bw = 5 v, 2 v, 8 bw = 10 w, bw = 7 z, 8 z, 10 bw = 7 Fig. 1. Example of constructing event-riven graph G(D) from a DTN graph D. In each ege, t enotes the time of the ege, an bw its banwith. For each noe, buf enotes the buffer size. In G(D), the vertical eges are intra-noe eges, while other eges are inter-noe eges. Note that the ege (u, w) is prune. Intra-noe Eges: For each set V v V \{s, }, sort the noes in ascening orer accoring to their corresponing times; namely V v = ( v, t 0, v, t 1,... v, t l ), where t 0 t 1... t l. For each i {0,..., l 1}, a the ege ( v, t i, v, t i+1 ) banwith buf(v) to E. Inter-noe Eges: For every e = (u, v) E, if u, t(e) V an v, t(e) V, a the ege ( u, t(e), v, t(e) ) to E. The banwith of the ege is bw(e). The event-riven graph is not a time-epenent graph, so all its eges exist all the time. Furthermore, a simple optimization can prune noes of v, t V for which there is no outgoing contact from super-noe v after time t. We next exten the construction to unirecte DTN graphs. A simple approach to hanle unirecte DTN graph coul have been treating each unirecte contact ege as two antisymmetric irecte eges occurring at the same time; however, this will result in a DTN graph for which the time function t : V E is not injective as require. We take a more refine approach in which we characterize each event not only by its noe-time pair, but also by its role (namely, sening or receiving event) an treating a sening event at time t at noe v as if it occurs just before the receiving event at time t at noe v. This implies that we assume that when a noe u sens ata to a noe v, it immeiately evacuates its buffer an therefore can simultaneously receive an store the same amount of ata from v. Fig. 2 illustrates how to convert a single unirecte contact to its corresponing event-riven sub-graph; notice that each noe in the event-riven graph is represente by a triplet corresponing to the super-noe, time an role (0 for sening event an 1 for receiving event). The formal efinitions are omitte from this paper ue to space consierations. We procee by proviing the following properties of eventriven graphs: roperty 1: If D = V, E is a irecte DTN graph, then the number of noes of its corresponing event-riven graph is at most 2 E an the number of eges is boune by E. If D = V, E is unirecte DTN graph, then the number of noes is boune by 4 E an the number of eges is at most 6 E. roperty 2: For any DTN graph D (irecte or unirecte), the event riven graph G(D) is a irecte acyclic graph with maximum in-egree 2, out-egree of 2 an total-egree of. roperty : The construction of an event-riven graph G(D) from a DTN graph D = V, E takes O( E ) time an requires O( E ) space. Finally, we note that the above-mentione construction oes not eal with the energy constraints of the DTN. A iscussion on these constraints an how to incorporate them in the eventriven graph will be presente in Section IV-C. IV. ROUTING AND SCHEDULING ALGORITHMS We emonstrate how to use the event-riven graph in orer to erive routing algorithms for DTNs. The algorithms will iffer from each other in their objective function. A. Minimum Delay (Work-span) ath In this section, the goal is to fin a feasible routing scheme path with strictly positive banwith for which t( ) is minimal (that is, its last contact e has the minimal possible time value of associate with it). Since the only requirement is that the banwith will be strictly positive, it is easy to show that it is sufficient to look at routing schemes which consist on only a single path = {p}. In [2], the authors gave a tailoremae moification to Dijkstra s algorithm in orer to solve this problem irectly on DTN graph. We will take a ifferent approach an solve the problem using the event-riven graph. The path can be foun by aing a global source noe s to G(D) an connect it with all noes belonging to supernoe s. Then, by performing any graph traversal algorithm e.g., breath first search (BFS) from noe s, one can easily compute which noes are reachable from s. Let R V be the noes that are reachable from s an belongs to supernoe, an let t be the minimum time associate with noes in R. Thus, the minimum elay is t an the minimum elay path is the path from s to, t as was iscovere in the graph traversal (for example, the path from s to, t in the BFS tree). The time an space complexity of this algorithm is O( E ). Minimum elay an minimum number of hops: Sometimes, in orer to reuce the number of transmissions, it is esirable to choose, among all minimum elay paths, the one with the minimum number of hops. This can easily achieve by first

5 computing the minimum elay t, then pruning G(D) so that all eges an noes with time more than t are elete, an finally running Dijkstra s shortest path algorithm on the prune G(D) giving each inter-ege a length of 1 an each intra-ege a length of 0. B. Maximum Banwith When the goal is to fin a feasible routing scheme that maximizes the total amount of ata that can be sent from s to, one can use a maximum flow algorithm on the event-riven graph. This stems from the following theorem, showing the correlation between a feasible routing scheme on D an a flow on G(D) (proof omitte); in the sequel we will assume that there is a global source s in G(D) which is connecte with infinite banwith eges to all the noes that belongs to super-noe s an in aition there is a global sink connecte with infinite banwith eges from all the noes that belong to super-noe. Theorem 4: Assume that a DTN graph D = V, E has no energy constraints (that is, for every v V, en(v) = + ). Then, there is an s flow on G(D) = V, E with value x if an only if there is a feasible routing scheme on D with the same value x. Moreover, the routing scheme on G can be efficiently constructe from the flow on G(D). The gist of the proof is an (efficient) construction of a legal flow on G(D) from each routing scheme on D, an vice versa. Basically, the flow on the inter-eges of G(D) is equal to the amount of ata sent on the contacts of G, whereas the flow on the intra-eges of G(D) correspons to the occupancy of D s buffer in a specific time interval. Moreover, it is important to notice that since Theorem 4 provies a metho to translate a given flow on G(D) to a routing scheme in D, the following result immeiately follows: Corollary 5: By applying a maximum-flow algorithm on G(D), one can efficiently fin a routing scheme on D with the maximum value. Maximum banwith with minimum elay: In many cases, it is esirable to choose, among all routing scheme with maximum banwith, the one that occurs the minimal ento-en elay an/or the one that requires the the least possible transmissions per byte of ata. Computing maximum flows on the event-riven graph, or subgraphs of it, is very useful for solving these problems as well. For example, Algorithm 1 shows how, by computing several maximum flows, one can fin a routing scheme with maximum banwith an minimum elay. The algorithm performs a binary search on the maximum flow values of subgraphs of G(D), prune accoring to the possible en times, until the maximum flow with minimum elay is foun; the number of iterations is boune by O(log E in ) where Ein is the set of incoming contacts (in the DTN graph) to the target noe. Other extensions: In orer to fin the routing scheme with maximum banwith an minimum number of transmission per byte, one can solve the maximum flow minimum cost problem, assigning a cost of 0 to each intra-ege an a cost of 1 to each inter-ege. These two extensions can be combine noen 1 Fining the maximum banwith routing scheme with minimum elay 1: G(D) CONSTRUCT-EVENT-DRIVEN-GRAH(D) 2: f MAX-FLOW(G(D)) f is a function from the eges of G(D) to their assigne flow : x e E in f(e) E in E is the set of incoming eges in G(D) to the global sink ; hence, x is the value of (max) flow f 4: T E in 5: SORT(T ) by ascening orer of eges times 6: l 0, h T 1 7: while l h o 8: m (l + h)/2 9: G (D) RUNE(G(D)) accoring to time T [m] f (e) 10: f MAX-FLOW(G(D )), x e E in 11: if x < x then 12: l m + 1 1: else 14: min elay T [m], h m 1 15: G (D) RUNE(G(D)) accoring to time min elay 16: f MAX-FLOW(G(D )) 17: CONSTRUCT-ROUTING-SCHEME-FROM-FLOW(f ) 18: return in orer to choose among the routing scheme with maximum banwith an minimum elay the one with minimum number of transmission per byte: we first apply Algorithm 1 to fin the minimum elay, then prune the graph accoringly an apply the maximum flow minimum cost algorithm. Alternatively, if the contact banwiths are integral (an therefore the flow is integral as well) one can scale the cost of the eges of E in accoring to their times, so that each minimum cost solution will immeiately yiel minimum elay; in this case the cost that shoul be assigne to inter-ege correspons to e E in is t(e) n x where n is the number of inter-eges in G(D) an x is the maximum flow on G(D). Another simple extension to the following algorithms is to fin routing schemes that achieve a certain banwith eman y, with minimum elay, minimum number of transmissions (hops), or both. This can be one, for example, by replacing the comparison in Line 11 of Algorithm 1 to be with y instea of x. Finally, when there are ifferent sources that shoul sen ata to ifferent estinations along the DTN, the problem can be moele as a multi-commoity flow problem on the corresponing event-riven graph. It is well-known that this problem can be solve using linear programming (in case fractional flows are allowe); many other results on this problem an its many variants exist in literature an can be irectly applie in this case as well (c.f. [16], [17] for more comprehensive surveys). In aition, one may use our construction to moel multicast scenarios in DTN (that is, when a single source shoul sen ata to many estinations) an tackle them using off-the-shelf algorithms available on time-inepenent networks (e.g. approximation algorithms for the minimum Steiner tree problem).

6 (t = 0, bw = 5) t = s u t = 1 v t = 4 (t = 2, bw = ) t = 5 s s, 0 5 Energy: 7 u, 0 v, 1 Fig.. Example of using a noe u as a transient buffer for another noe v. The banwith bw of all contact is unless otherwise specifie. s, 2 u, 1 v, 4 C. Dealing with Energy Constraints In this section we introuce the energy constraints on noes of the DTN graph D an iscuss how to evise algorithms taking into account these constraints as well. Recall that since each intermeiate noe (that is, a noe other than s an ) forwars all the ata it gets, the energy constraint translates to the amount of ata a noe receives over the entire time span. A key phenomenon, which makes hanling energy constraints more ifficult, is the usage of noes as transient buffers, such that the flow of ata may form cycles on the DTN graph (but never on the event riven graph). These cycles, in turn, yiel that, in orer to transmit x bytes from s an that go y times through super-noe v (i.e. in y cycles), the amount of energy neee at super noe v is x y. For comparison, in a (traitional) max-flow problem there is no incentive to employ cycles in the graph, an therefore such a problem oes not exist. Fig. epicts a scenario in which such a phenomenon happens: suppose that all noes but u o not have any buffer or energy constraints, an, on the other han, buf(u) = an en(u) = 7; furthermore enote by e i the contact for which t(e i ) = i. We first compute the maximum banwith routing scheme without taking into account the energy constraints: the resulting scheme is = {p 1, p 2 } of total banwith 6, where p 1 = (s, e 0, u, e 1, v, e 4, u, e 5, ) of banwith an p 2 = (s, e 2, u, e, ) of banwith. Note that p 1 contains a cycle (u is visite twice), an in fact noe v was use as a transient buffer for the ata of p 1 between time 1 an. The total amount of energy spent at noe u is 9, thus exceeing en(u). To correct this one shoul ecrease the banwith that goes through u; since each byte along p 1 is counte twice, it is sufficient to reuce the banwith of p 1 to 2, resulting in total banwith of 5. Going back to the event-riven graph G(D), recall that the energy constraints were not realize in its construction. The reason behin this is that energy is counte over the entire time span an not only between events; thus, it is common to all noes that belongs the same super-noe, as illustrate in Fig. 4. Unlike Section IV-B, where we apply combinatorial algorithms to solve maximum-flow relate problems, here we formalize the problem on the event-riven graph as a linear program. It is important to notice that since the resulting flow nee not be integral, solving the linear program yiels an efficient algorithm to fin the maximum flow. For example, the following linear program escribes the problem of fining a maximum flow on the graph, respecting both the buffer an energy constraints: u, 2 u, u, 4 u, 5 Fig. 4. The event-riven graph of the DTN epicte in Fig.. Non-labele eges have infinite banwith.,, 5 maximize f s such that f e bw(e) e E j:(j,i) E f ji j:(i,j) E f ij 0 i V k V v j:(j,k) E f jk I jk en(v) v V where I jk is an 0 1 inicator variable that equals 1 if an only if (j, k) is an inter-ege. Note that the ual of this program is e E bw(e) e + v V en(v)rv minimize such that ij p i + p j + r v(j) I ij 0 (i, j) E p p t 1 e, r v 0 where v(j) is the super-noe v V that the noe j V belongs to. It is important to notice that unlike a traitional MAXIMUM FLOW problem, the solution of the ual problem is not necessarily integral, implying that a max-flow min-cut theorem is unlikely to hol. In fact, the example epicte in Fig. shows also that a simple augmentation-path base algorithm cannot solve this problem, since one can first choose p 1 with banwith without violating the constraints. V. ERIODIC DTN In some cases the DTN graph escribes a perioic pattern of contact between the noes. Specifically, suppose the DTN perio length is given by τ > max e E t(e), then if a contact between some noe u an another noe v has time t(e), it implies that u can sen ata to v at times t(e) + kτ for any k N. Moreover, since the system is now long-live, we treat the energy constraints as the energy that can be spent by a noe over one perio. We focus on a maximum banwith problems where the goal is to eliver as much ata as possible per perio. Since we measure the banwith per perio, one can mistakenly assume that the problem can be solve by looking only on a single perio; however, as the example epicte in Fig. 5 shows, this is not the case: when looking on a single-perio

7 s t = 4 u t = v Fig. 5. Example of a perioic DTN which can eliver banwith of 1, while uring a single perio the DTN cannot eliver any ata. The banwith of all contact bw = 1. DTN, it is clear that no ata can be transferre from the source to the estination, however for a perioic DTN the amount of banwith per perio goes to 1 as ata is sent over the links at times kτ +4, (k +1)τ +, (k +2)τ +2, (k +)τ +1 for any perio k N. This scenario somehow resembles the pipeline paraigm. Therefore, we moel the problem by assuming a perioic routing scheme, which treats traffic transmitte from source s at time kτ + t as the traffic transmitte at time t. Then, we can solve the maximum banwith per perio problem by looking only on the traffic transmitte from the source at the first perio, while respecting the following aitional constraints: let e be a contact, an let f(e, t, ) be the flow assigne to contact e at time t uner routing scheme. Thus, k=0 f(e, t(e) + kτ, ) bw(e); namely, the total amount of flow on a contact over all perios oes not excee its banwith. recall that occ(v, t, ) is the number of bytes store at noe v at time t uner routing scheme. Thus, we require for all v V an t < τ, k=0 occ(v, t + kτ, ) buf(v). let recv(v, k, ) to be the number of bytes receive by noe v uring perio k uner routing scheme. Thus, we require for all v V that, k=0 recv(v, k, ) en(v). Hence, perioic DTN graph can be conceptually reuce to a cyclic event-riven graph G(D) = V, E in which for each super-noe v V there is an extra ege of banwith buf(v) connecting the last noe (orere by time) of V v with the first noe of V v. Running a maximum-flow algorithm on this event-riven graph will give the maximum banwith per perio that can achieve. Moreover, given the maximum flow f, one can construct a routing scheme (using some path splitting algorithm) an compute the corresponing elay of by counting the number of extra eges in each path. Let k max be the maximum number of perios neee to complete any path in (that is, the number of extra eges in the path plus one). It is important to notice that k max τ is only an upper boun on the minimum elay neee to achieve the maximum banwith. In fact, computing the routing scheme with minimum banwith an maximum elay is more complicate in case the DTN is perioic. This is one by consiering an eventriven graph G(D) = Ṽ, Ẽ which hols k max copies of the original event-riven graph G(D) = V, E. Noes corresponing to super-noe s appears in G(D) only in the first copy (implying that we consier only traffic transmitte from the source at the first perio) an the last noe (orere by time) of some super-noe v at k-th copy is connecte by t = 2 w t = 1 an intra-ege of banwith buf(v) to the first noe of v at k +1-th copy; the set of all newly-ae eges between copies k an k + 1 is enote by E k,k+1. Furthermore, let the set of all noes that belongs to super noe v V in the k-th copy be Ṽ v,k, while the set of all k max copies of an ege e E E 0,1 is enote by Ẽe. Finally, we assume that the global source noe s is connecte to all noes that belongs to super noe s while the global sink noe is connecte to all noes that belongs to super noe in all k max copies. The aitional constraints of the perioic DTN can then be realize by aing the corresponing linear inequality constraints, resulting in the following linear program that compute the maximum-flow on G(D): maximize f s such that ẽ Ẽe fẽ bw(e) e E E 0,1 j:(j,i) Ẽ fji j:(i,j) Ẽ fij 0 i Ṽ fjiiji en(v) v V kmax k=0 i Ṽv,k j:(j,i) Ẽ where I ji is an 0 1 inicator variable that equals 1 if an only if (j, i) is an inter-ege. Finally, this proceure can be plugge into Algorithm 1, thus computing the maximum flow with minimum elay. VI. EXERIMENTAL RESULTS Many experimental traces, publicly available in [18], store the contacts observe in ifferent mobility scenarios (like, public transportation systems, humans); they allow to buil irectly the DTN graph, which is the input of our optimization framework. For lack of space, we have chosen to show just some results referre to the mobility of a set of 7 buses running routes for 16 weeks in the UMass campus []. These buses operate aily as ictate by their time scheules but also by their failures an maintenance stops. The trace provies basically the sequence of all the contacts in the following format: time, source bus, estination bus, capacity of the contact (measure in bytes). We have consiere just the contacts referre to the thir week of the experiment. We have run the optimal algorithm to maximize the banwith in a ata flow between two given buses. Fig. 6 shows the curves of the minimum elay necessary to transfer a given amount of ata for two cases, corresponing to two specific pairs of source an estination buses (similar results were observe for all the other possible pairs). Both curves are step functions. Inee, assume that, for a specific value of banwith x an elay y, the last contact in the routing (with contact time y) has not been yet completely saturate. When x increases, then the elay remains the same until the banwith of the last contact is fully exploite. At that point, the elay jumps to the time of the next contact exploite. The graph can be seen also as the representation of the capacity-elay region achievable in the two cases. Inee, there is no algorithm that is able to achieve any point below an to the right of each curve, since the routing an scheuling scheme are optimal from both the capacity an the elay

8 Delay [hours] Bus pair Bus pair Bus pair region Bus pair region Capacity [Mbytes] Fig. 6. Minimum elay for two specific pairs of buses. Curves efine also the optimal elay-capacity regions for the two cases. Capacity [Mbytes] Buffer constraine - perioic Energy constraine - perioic Buffer constraine Energy constraine Energy or Buffer [Mbytes] Fig. 7. Maximum banwith achievable for the bus pair as function of the buffer an energy available at each bus point of view. Note that this region shows some performance limitations of the DTN consiere in the experiment; this is coherent with previous work [15] an ue to the fact that time scheule for public transportation is inherently esigne to reuce contacts among the buses. Fig. 7 consiers only the ata flowing from bus 046 to 041, an shows the effect of the constraints on the resources, i.e. the energy an the buffer available at each noe; for simplicity, we assume an homogeneous network in which all the noes share the same amount of buffer an have the same amount of energy available (measure as the maximum number of bytes that can be transmitte by a noe other than the source). As expecte, capacity is a monotonic function of the energy an the buffer, an its maximum value (80 Mbytes, coherently with Fig. 6) is obtaine for sufficiently large energy an buffer. In aition, the constraint on the energy tens to affect more severely the performance, since the buffer, as a resource, can be re-utilize in ifferent times, whereas the energy can be exploite only once. When the buffer is very small, the routing is so constraine that the buffer tens to be exploite just once an the effects of the energy an the buffer are equivalent. Furthermore, Fig. 7 shows the effect of consiering perioic DTNs. The qualitatively behavior is similar as before. erioic DTNs cannot benefit from small value of resources, especially from small values of energy, which can be exploite only once. Larger values of resources allow to achieve larger capacity, an the maximum possible capacity correspons, in the original DTN graph, to the minimum between the cut aroun the source noe an the cut aroun the estination noe. The capacity gain by consiering perioic DTNs is more than 50% for buffer/energy larger than 1 Mbytes, an reaches 122% when resources are unboune. VII. CONCLUSIONS We have propose an optimization framework to fin the optimal routing for eterministic DTNs. Our novel approach is very general an base on a construction of an eventriven graph from the classical DTN graph; this allows us to exploit stanar algorithms from Graph Theory. We have emonstrate our approach by showing solutions to ifferent cost functions (maximum banwith, minimum elay) an constraints (maximum buffer an energy per noe). We have also iscusse some numerical results referring to a real mobile network, whose DTN graph is known through traces. REFERENCES [1] M. Grossglauser an D. Tse, Mobility increases the capacity of a hoc wireless networks, IEEE/ACM Trans. Netw., vol. 10, no. 4, pp , [2] S. Jain, K. R. Fall, an R. K. atra, Routing in a elay tolerant network, in ACM SIGCOMM, 2004, pp [] C. Liu an J. Wu, Scalable routing in elay tolerant networks, in ACM MobiHoc, 2007, pp [4], Routing in a cyclic mobispace, in ACM MobiHoc, 2008, pp [5] Delay tolerant networking research group. [Online]. Available: [6] J. Burgess, B. Gallagher, D. Jensen, an B. N. Levine, Maxprop: Routing for vehicle-base isruption-tolerant networks, in INFOCOM, [7]. Hui, A. Chaintreau, J. Scott, R. Gass, J. Crowcroft, an C. Diot, ocket switche networks an human mobility in conference environments, in ACM WDTN, 2005, pp [8] M. C. Gonzalez, C. A. Hialgo, an A.-L. Barabasi, Unerstaning iniviual human mobility patterns, Nature, vol. 45, no. 7196, pp , [9] Crossbow technology, inc. [Online]. Available: [10] A. Balasubramani, B. N. Levine, an A. Venkataramani, DTN Routing as a Resource Allocation roblem, in ACM SIGCOMM, [11] S. Merugu, M. Ammar, an E. Zegura, Routing in space an time in networks with preictable mobility, Georgia Institute of Technology, Tech. Rep., [12] L. R. For an D. R. Fulkerson, Constructing maximal ynamic flow from static flows, Operation Research, vol. 6, pp , [1] B. Hoppe, Efficient ynamic network flow algorithms, h.d. issertation, Cornell University, [14] M. Skutella, An introuction to network flows over time, in Research Trens in Combinatorial Optimization. Springer, 2008, pp [15] A. Di Nicolò an. Giaccone, erformance limits of real elay tolerant networks, in IEEE WONS, 2008, pp [16] B. Awerbuch an F. T. Leighton, Multicommoity flows: A survey of recent research, in ISAAC, 199, pp [17] G. Karakostas, Faster approximation schemes for fractional multicommoity flow problems, in ACM-SIAM SODA, 2002, pp [18] Crawa repository. [Online]. Available: eu