QoS and Sensible Routing Decisions

Transcription

1 QoS and Sensible Routing Decisions Erol Gelenbe Dept. of Electrical & Electronic Engineering Iperial College London SW7 2BT Abstract Network Quality of Service (QoS) criteria of interest include conventional etrics such as throughput, delay, loss, and itter, as well as new QoS criteria based on power utilization, reliability and security. In this paper we suggest a theoretical fraework for the characterization and coparison of adaptive routing algoriths which use QoS as the criterion to select between different paths that data transitted in the network (e.g. packets, connections, etc.) ay take. Our obective is not to analyze QoS, but rather to design randoized routing policies which can iprove QoS. We define a QoS etric as a non-negative rando variables associated with network paths which satisfy a sub-additivity condition along each path. We define the QoS of a path, under soe routing policy, as the expected value of a non-decreasing easurable function of the QoS etric. We discuss sensitive and insensitive QoS etrics, the latter being dependent on the routing policy which is used. We describe routing policies siply as probabilistic choices aong all possible paths fro soe source to soe given destination. Sensible routing policies are then introduced: they take decisions based siply on the QoS of each available path. Sensible policies, which ake decisions based on the QoS of the paths, are introduced. We prove that the routing probability of a sensible policy can always be uniquely obtained. A hierarchy of -sensible probabilistic routing policies is then introduced and we provide conditions under which an + sensible policy provides better QoS on the average than an sensible policy. Keywords: Matheatics of Networks, Quality of Service, Routing. Introduction Quality of Service (QoS) has now becoe a central issue in network design, and there is a vast and significant literature on the proble of estiating certain specific quality of service paraeters (e.g. loss or delay) for given traffic characteristics and a given network topology [3, 4]. Typically such work has considered single buffer odels (finite or infinite), or odels of cascaded nodes with or without interfering traffic. There has also been uch work on schees for obtaining better QoS through routing [6, 4, 5], on scheduling techniques in routers to achieve desired QoS obectives [3], as well as on the analysis of QoS resulting fro the detailed behavior of protocols such as TCP/IP. This work is supported by a grant fro EPSRC. P2/

2 The ixed wired and wireless network topologies that are becoing coon, including fixed and ad-hoc connections, create the need to rationally exploit dynaically variable routing as a function of network conditions, since the a pplications that use such networks have QoS requireents such as delay, loss or itter, as well as reliability and low power utilization. In recent years we have conducted research on routing algoriths with two quite different applications in ind: routing in robotic navigation, and routing algoriths in wired networks and wireless ad-hoc networks. The applied research we are currently conducting basically addresses adaptive routing algoriths in a discrete structure. In robotic navigation, the discrete structure is a discrete grid of points in a terrain (i.e. a terrain database ). In a counication network the discrete structure is obviously the set of nodes in a Mobile Ad-hoc Network or in a wired network, and routing decisions are based on optiizing QoS obectives. The QoS etrics used to route robots in a terrain are based on iniizing travel ties to destination, iniizing the probability of being destroyed by adversaries (which ay be fixed or obile), iniizing power utilization due to otion, and the obvious need to avoid obstacles. In an ad-hoc network, QoS requireents include the iniization of packet loss, the iniization of end-to-end delay, the iniization of overhead, the iniization of power consuption for routing purposes, or a cobination of soe of these criteria. Motivated by our prior work on adaptive network routing algoriths [7, 9, 2, 0, ], in this paper we investigate soe basic atheatical probles concerning QoS driven routing. The ai of this work is not to analyze QoS, but rather to show that certain randoized routine policies can iprove QoS. We define QoS etrics as non-negative rando variables associated with network paths which satisfy a sub-additivity condition along each path. We then describe routing policies siply as probabilistic choices aong all possible paths fro soe source to soe destination. Increental routing policies are defined as those which can be derived fro independent decisions along each sub-path. We define the QoS of a path, under soe routing policy, as the expected value of a easurable function of the QoS etric. We discuss sensitive and insensitive QoS etrics, the latter being dependent on the routing policy which is used. Sensible routing policies are then introduced; these policies take decisions based siply on the QoS of each allowable path. Finally, a hierarchy of -sensible probabilistic routing algoriths is introduced. The 0- sensible ruting policy is siply a rando choice of routes with equal probability, while the -sensible policy uses the relative QoS for each alternate route to ake select a path. An - sensible policy uses the th power of the QoS for each alternate path, rather than ust the st power. Thus it siply uses the sae inforation in a different anner. It is particularly interesting that we can prove that an + -sensible policy provides better resulting average QoS than an -sensible policy, provided that the QoS etric is insensitive. We also prove that under certain sufficient conditions, the sae result holds for senstive QoS etrics. P2/2

3 . Quality of Service (QoS) etrics A QoS etric relates to soe specific data unit, the ost obvious exaple being a packet. However ore broadly, a data unit ay be a significant sequence of packets which belong to the sae connection. A QoS etric q can be illustrated by the following exaples: q D ay be the delay D experienced by a packet as it traverses soe path in the network, or It ay be the binary variable q r = [the path is connected], or q LR = L/n ay be the nuber of packets lost L divided by the nuber of packets sent n (i.e. the packet loss rate) for a sequence of packets, or q ay be the average itter experienced by n successive packets: q J = n n l=2 (R l R l ) (S l S l ), () where S l is the date at which packet l was sent fro the source, and R l is the tie at which packet l arrives to its destination, etc., or q ay be the nuber of hops a packet has to traverse, or q ay be the power expended by the network nodes to service and forward a packet as it travels through a path in the network, or q ay be the effective delay obtained by coposing soe of these values, such as: q = ( q LR )q D + q LR (T o + q D ), = q D + q LR T o. where T o is the (large) tie-out delay which triggers a packet s retransission. Soe paths ay not be connected because physical links ay not exist. Also since we are dealing with potentially unreliable wired networks as well as ad-hoc networks, soe or all links in the path ay be connected only with soe probability. If a link along a path V is disconnected, we can still copute quality of service etrics, for instance we ay then have q D (V ) = +, q LR (V ) =, and q J (V ) = +..2 Routing policies Let the nodes in a network be denoted by a fine set of natural nubers {0,,2,... N}. Definition A path in the network starting at node i and ending at node v d is denoted by V i = (i,...,v d ). It is a sequence of nodes such that the first node is i, the last node is v d, and no node in the sequence V i appears ore than once. We associate QoS etrics with paths in the network. Let FV i (v d ) = {Vi,V i 2,...,V i } be the set of all distinct, but not necessarily disoint, paths fro node i to node v d in the network. P2/3

4 Definition A routing policy for source-destination pair (i,v d ) is a probability distribution π FV i(v d ) on the set FV i (v d ), that selects path V i FV i (v d ) with probability π FV i(v d ) (V i ) for each individual data unit which is sent fro node i to node v d. For any V i FV i (v d ), we ay write V i = (i,...,l,n,...,v d ) as a concatenation of a prefix path and a suffix path: V i = P p i.ss n where P p i = (i,...,l), Sn s = (n,...,v d ). Consider now the sets of paths fro i to l, FV i (l) and fro n to v d, FV n (v d ). Whenever needed, Π will denote the routing policy for the network as a whole, i.e. the set of rules that assign unique paths for each data unit oving fro any source node to any destination in the network. F V will denote the set of all paths fro all possible source to destination nodes in the network. 2 QoS etrics Definition A QoS etric for path V is a rando variable q Π (V ) which takes values in {0,+ }, such that for V = V.V 2 (i.e. V is coposed of path V followed by path V 2 ): q Π (V ) q Π (V ) + q Π (V 2 ), a.s. Note that the requireent that the QoS etric be sub-additive covers any strictly additive etrics of interest such as packet or cell loss rates, delay, path length (nuber of hops), and power dissipation. Other etrics such as path reliability and available bandwidth are subadditive, and are also covered by our definition. For a path V coposed of two successive sub-paths V = V.V 2, the following are obviously sub-additive: q available BW (V ) = inf(q available BW (V ),q available BW (V 2 )) a.s., q available BW (V ) + q available BW (V 2 ) a.s., q r (V ) = [q r (V ) and q r (V 2 )] a.s., q r (V ) + q r (V 2 ) a.s. where q r (.) is treated as a logical binary rando value in the third equation, and as a nuerical (binary) rando value in the last equation. 2. QoS etrics and QoS In the sequel q πf V i (v d ) (V i ) will denote the QoS etric q easured on path Vi, when the policy π FV i(v d ) is applied to data units travelling fro node i to v d using the set of paths FV i (v d ), V i FV i (v d ). We soeties write q with a subscript, e.g. q πf V i (v d ) D (V i ) or V qπf i (v d ) LR (V i ) to indicate that it designates soe specific etric such as packet delay or packet loss rate. Definition Let u be a non-decreasing easurable function and q be a QoS etric. The QoS for data units sent on the path V i using policy π FV i(v d ), fro source i to destination v d along the set of paths FV i (v d ) is siply the expected value E[u(q πf V i (v d ) (V i ))], i.e. the expected value of a easurable function of a QoS etric. The reason for assuing that u is an increasing function (i.e., non-decreasing) is that we want to the QoS to reflect the trend of the QoS etric. If the QoS etric has a larger value P2/4

5 reflecting soe degradation in the path, we want the QoS also to reflect this degradation, or at least not to reflect an iproveent. Let us illustrate this with an exaple: let the QoS etric q πf V i (v d ) D (V i )) be the path delay, and let u(.) = (.) be the indicator or characteristic function. The QoS E[(q πf V i (v d ) D (V i )) > T)] = Prob[qπF V i (v d ) is the probability that the path delay is larger than T. D (V i )) > T] 2.2 Sensitive QoS etrics The value for soe path of a routing sensitive, or siply sensitive QoS etric q increases when the probability of directing traffic into that path increases; exaples include path delay and path loss ratio. An exaple of an insensitive QoS etric is the nuber of hops along a path; the power dissipated per data unit on a path ay also be insensitive. Even when the probability of sending traffic down a given path is zero, we ay assue that the path can be infrequently tested to obtain the value of the QoS etric of that path, or the path QoS ay be known via prior inforation (e.g., available bandwidth, nuber of hops, or the path s power dissipation). Definition We will say that the QoS etric q is sensitive on the set FV i (v d ), if for any two routing policies π FV i(v d ) and π FV i(v d ) and any path V i FV i (v d ): { π FV i(v d ) (V i ) > π FV i(v d ) (V i ) } V Prob[qπF i (v d ) (V i ) > x] V Prob[qπ F i (v d ) (V i ) > x], for all x > 0. We say that q is insensitive on the set FV i (v d ) if for any path V i, and any two routing policies such that π FV i(v d ) (V i ) π FV i(v d ) (V i ): Prob[q πf V i (v d ) (V i ) > x] = V Prob[qπ F i (v d ) (V i )], for all x > 0. 3 Sensible Routing Policies A sensible routing policy is one which: Selects paths only using the expected value of the QoS, i.e. the expected value of a function u of the QoS etric q for each path, as the criterion for selecting the probability that a path is chosen, Selects the path for a new data unit independently of the decision taken for the previous data unit. The practical otivation for considering sensible routing policies is that () averages of QoS etrics, or of functions of QoS etrics, are typically easy to estiate, and (2) decisions which are successively independent for successive data units are easier to ipleent. P2/5

6 Definition Let u be a non-decreasing easurable function. A sensible routing policy (SRP) fro node i to destination v d based on the QoS etric q is a probability distribution π FV i(v d ) on the set FV i (v d ) such that: π FV i(v d ) (V i ) = f i (E[u(qπF V i (v d ) (V i ),u(q πf V i (v d ) (V 2 i ),...,u(q πf V i (v d ) (V FV i(v d ) ))]), (2) for a function f i : R [0,], for each V i V i FV i (v d ) FV i (v d ), such that: π FV i(v d ) (V i ) =, (3) and for each path V i, the function f i (y,...,y,...,y FVi (v d ) ) defined in (2) is strictly decreasing in its arguent y, with li y + f i (y,...,y FVi (v d ) ) = 0. Coent Thus a SRP is a routing policy which decides on routing based only on the QoS of each path, such that whenever the value of the QoS for any path increases then the probability of selecting that path decreases. Exaple A siple exaple of a SRP is the following: π FV i(v d ) (V i ) = E[q πf V i (v d ) D all s (V i )] E[q πf V i (v d ) D (V s i )], (4) which says that packets are directed to the paths with a probability which is inversely proportional to the average delay. In a sensible routing policy, the probability that a specific path is selected will depend on the QoS of that path, which in general depends on the policy itself, i.e. on the probability that the path is selected, unless the policy is insensitive. Thus there is the question of whether we are able to copute the routing probabilities. The following theore provides sufficient conditions for being able to do this. Theore If π FV i(v d ) is a sensible routing policy on FV i (v d ), then the solution to (2) exists and is unique for each path V i. Proof For any path V i, consider: the function f i (y,...,y,...,y FVi (v d ) ) of equation (2), which (strictly) decreases when y increases, and the path QoS y (π FV i(v d ) ) = E[u(q πf V i (v d ) (V i )]. Since { π FV i(v d ) (V i ) > π FV i(v d ) (V i ) } Prob[qπF V i (v d ) (V i ) > x] Prob[qπ F V i (v d ) (V i ) > x], (5) for all x > 0, the path QoS is an increasing function (not strictly) y (π) of its arguent, the probability π, because of (5), and because u is an increasing function. P2/6

7 Thus the solution of equation (2) for any V i is obtained at the intersection of a non-negative, strictly decreasing function f i of y which tends to zero, and an increasing non-negative function y of f i. 4 -Sensible routing policies (-SRP) In this section we extend the concept of a sensible policy to ore sophisticated usage of QoS to ake routing decisions. We construct a hierarchy of -sensible policies, where the -sensible policy is ust the sensitive policy defined earlier, and the 0-sensiblene policy is a rando uninfored choice between paths with equal probability. What is particularly interesting is that, ust by increasing the value of we are guaranteed to achieve better overall QoS, when the QoS etric is insensitive. The sae result can be obtained in the sensitive case as well under certain sufficient conditions. Definition For a natural nuber, an -sensible routing policy ( SRP) fro node i to destination v d based on the QoS etric q is a probability distribution π FV i(v d ) on the set FV i (v d ) such that: π FV i(v d ) (V i ) = E[(u(q π(f V i (v d ) (V all s i )) ] E[u(q πf V i (v d ) (Vi s)) ]. (6) We will use the notation π SRP[FV i(v d )] to denote the fact that the policy π on the set of paths FV i (v d ) is sensible, and the corresponding QoS value will be denoted by q π SRP[F V i (v d )] (Vi s ) for path Vi. Note that a 0 SRP is ust a rando choice aong paths, with equal probability. 4. The -sensible routing theore when the QoS etric is insensitive In this section we assue that q is insensitive on the set FV i (v d ), and consider -SRP routing policies as defined in (6). To siplify the notation, let us associate the index with the path V i and write: W () = E[u(q π SRP[F V i (v d )] (V i ))]. (7) When q is insensitive, we will siply write W. Using (6) and (7) we have: Q π SRP[V F i ] = We first prove the following siple result. = W W = W. (8) Lea For any W 0, W k 0, 2 (W 2 + W k ) + W W k P2/7

8 or W W k + W k W 2. Proof Since (W W k ) 2 0, we have (W 2 + W 2 k ) 2W W k, and therefore (W + W k ) 2 4W W k, or: 2 (W 2 + W k ) + W W k and therefore: which can be written as: copleting the proof. (W + W k )( W + W k ) ( W k W + W W k ) 4 We will call the following result the -SRP theore (-sensible routing theore) for insensitive etrics. Theore 2 If q is insensitive on the set FV i (v d ), the policy (+)-SRP is better than -SRP for, i.e.: Q π SRP[V F i ] Q π(+) SRP[V F i ]. Proof Fro Lea, we have that for any W 0, W k 0, W W k + W k W 2, and ultiplying both sides by /(W W k ) we obtain: W W + k + W + W k 2 W i W. Suing for,k =,...,n and adding identical ters on both sides, we have: = (W + )2,k=; k { W W + k + W + W k } = (W )2 +,k=; k 2 W W k, or ( = W )( = W + ) ( = W ) 2, This can be written as: P2/8

9 = W n = W = = W W +, or in the final for: = = W W W = = W W + W +, which copletes the proof. Q.E.D. It is obvious that for an insensitive QoS etric, selecting to be very large is good, since this will lead to choosing the path with the best QoS if such a path exists. We suarize this point in the following reark. However, if the QoS etric is sensitive then the atter is quite different, as will be discussed in the next section. Reark 3 Suppose that q is insensitive on the set FV i (v d ), and that path Vi following sense: is best in the W < W 2,... W n. Then li Q π SRP[V F i ] = W. Proof Using: Q π SRP[V F i ] = = W W = W which yields the result when we take.q.e.d., (9) = W + W ( W W ) + (W W ), (0) 4.2 The sensible routing theore for sensitive QoS etrics When the QoS etric is sensitive, the QoS varies with the load on the paths. This is of course the ost coon situation in practice, e.g. for QoS etrics such as delay, packet or cell loss, etc.. Thus we cannot generalize Theore 2 to the case where the QoS etric is sensitive. However we can provide necessary and sufficient conditions which will yield a siilar result. Let δ be defined as follows: δ = n = n (W ( + )) (W ()). () This leads to the sensible routing theore for sensitive QoS etrics. = Theore 4 If q is sensitive on the set FV i (v d ), the policy ( + )-SRP is better than -SRP for 0: P2/9

10 provided that the following condition holds: Q π SRP[V F i ] Q π(+) SRP[V F i ], [ = (W (+)) = (W ()) ] = (W (+)) + [ = (W (+)) = (W ()) ] = (W (+)). Proof Fro Theore, we know that for any W 0, =,... n, = W n = W = = W W +, and in particular this is true if we set W = W ( + ), so that = (W (+)) n = (W (+) = = (W (+)), (W (+)) + or: [ = (W (+)) ][ = (W (+)) + ] [ = (W (+)) ][ = (W (+)) ]. Using the definition of δ we have: [ i= (W()) + δ ][ = (W (+)) + ] [ = (W ()) + δ ][ = (W (+)) ], so that: [ = (W ()) ][ = (W (+)) + ] = (W ()) ][ = (W (+)) ] δ [ = (W (+)) ] δ [ = (W (+)) + ]. Condition (??) is therefore sufficient since it iplies that: δ [ = (W (+)) ] δ [ = (W (+)) + ] 0, which copletes the proof. Q.E.D. 5 Conclusions In this paper we suggest a theory of routing based on QoS. We have distinguished between QoS etrics, and QoS. Variable and adaptive routing have again becoe of interest in networking because of the increasing iportance of obile ad-hoc networks. In this paper we have developed a fraework for the study of adaptive routing algoriths which use the expected QoS to select paths to their destination. Our obective is not to analyze QoS, but rather to design randoized routing policies which can iprove QoS. P2/0

11 We define QoS etrics as non-negative rando variables associated with network paths that satisfy a sub-additivity condition along each path. We define the QoS of a path as the expected value of a non-decreasing easurable function of the QoS etric. We discuss sensitive and insensitive QoS etrics, the latter being dependent on the routing policy which is used. An exaple of an insensitive QoS etric is the nuber of hops on a path, since it will not change with the fact that this particular path is selected by the route selection. We describe routing policies as probabilistic choices aong all possible paths fro soe source to soe given destination. Sensible routing policies are then introduced: they take decisions based siply on the QoS of each possible path. Sensible policies, which ake decisions based on the QoS of the paths, are introduced. We prove that the routing probability of a sensible policy can always be uniquely deterined. A hierarchy of -sensible probabilistic routing policies is then introduced. A 0 sensible policy is siply a rando choice of routes with equal probability, while a sensible policy selects a path with a probability which is inversely proportional to the (expected) QoS of the path. We prove that an + sensible policy provides better QoS on the average than an sensible policy, if the QoS etric is insensitive. We also show that under certain conditions, the sae result also holds for sensitive QoS etrics. In future work we will consider yopic policies which only exaine partial inforation based on the QoS of initial portions of the possible paths in order to ake decisions. We will also consider Increental routing policies which can be derived fro independent decisions taken at certain points (or nodes) along paths. Finally, we plan to exploit the sub-additivity of the QoS etrics to prove asyptotic properties for very large networks. References [] D. Bertsekas and R, Gallagher Data Networks, Prentice Hall, Englewood Cliffs, NJ [2] H. Ahadi, R. Guerin and M. Nagshineh Equivalent capacity and its application to bandwidth allocation in high speed networks, IEEE J. on Sel. Areas in Co., Vol. 9, , 99. [3] E. Gelenbe, X. Mang and Y. Feng Diffusion cell loss estiates for ATM with ulticlass bursty traffic, Coputer Systes Science and Engineering, Special Issue: ATM Networks, Vol., No. 6, pp , Noveber 996. [4] V. Srinivasan, A. Ghanwani, E. Gelenbe Block cell loss reduction in ATM systes, Coputer Counications, Vol. 9, pp , 996. [5] D. Minoli and E. Minoli, Delivering Voice over IP Network, John Wiley & Sons, New York, 998. [6] S. Chen and K. Nahrstedt, An overview of Quality-of-Service routing for the next generation high-speed networks: Probles and Solutions, Network Magazine, Nov/Dec [7] E. Gelenbe, E. Seref, Z. Xu Towards networks with intelligent packets, Proc. IEEE-ICTAI Conference on Tools for Artificial Intelligence, Chicago, Noveber 9-, 999. [8] E. Gelenbe, Zhi-Hong Mao and Y. Da-Li (999) Function approxiation with spiked rando networks IEEE Trans. on Neural Networks, Vol. 0, No., pp. 3 9, 999. P2/

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