1 Introduction Recently there has been increased interest in providing real-time services over Internet. To this eect, IETF has dened two kinds of qua

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1 Call Admission and Resource Reservation for Guaranteed QoS services in Internet S. Verma a;1, R. K. Pankaj a and A. eon-garcia a a Network Architecture aboratory, Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (sanjeev, pankaj, alg)@comm.utoronto.ca Abstract Many audio and video \play-back" applications require that all of their packets arrive within their play-back time. Furthermore, the applications with the hard real-time requirements also require guaranteed service from the network. These applications are covered under the \Guaranteed Quality of Service" specications by the IETF. It guarantees that the packets will arrive within the guaranteed delivery time and the packets will not be discarded due to queue overows, provided the ow trac stays within its specied trac parameters. This paper gives an ecient and distributed algorithm based on the cost function to divide the end-to-end guaranteed QoS requirements into local QoS requirements which are then mapped into local resource requirements. The algorithm operates in three phases. In the rst phase, the nodes in the selected route give the information about their utilization level and the parameters associated with the scheduling algorithm used by them. The receiving node, based upon the information provided by the nodes in the route and its end-to-end delay requirements, takes the call admission decision. Once the decision to admit the call is taken, the nodes in the route do the resource allocation to meet the delay bound. The sender node now calculates the slack produced if any. The slack is then distributed to the nodes in the third phase. The results obtained are then shown to be applicable to the resource allocation for multicast connection establishment. Key words: Internet, PGPS, Guaranteed Quality Of Service, Resource Allocation and Optimization, Admission Control. 1 This work was supported in part by grants from Nortel, Canadian National Science and Engineering Research Council and Canadian Commonwealth Fellowship Preprint submitted to Elsevier Preprint 19 December 1997

2 1 Introduction Recently there has been increased interest in providing real-time services over Internet. To this eect, IETF has dened two kinds of quality of service (QoS) for dierent real-time applications: Guaranteed QoS and Controlled oad QoS [1]. The network elements conforming to these specications provide different end-to-end delay behaviour. The guaranteed QoS service provides rm end-to-end delay bound and no queuing loss for conforming packets of a data ow. On the other hand Controlled oad QoS service does not provide any rm quantitative guarantees. If the ow is accepted for Controlled oad QoS service, the network elements in the path make a commitment to oer the ow a service equivalent to that seen by a best-eort ow on a lightly loaded network. In this paper, we consider the call admission and resource reservation problems for the applications that need guaranteed QoS. The connection set-up process consists of two phases. In the rst phase, a route is established from the source to the destination. The second phase consists of checking whether or not the application can be admitted on the chosen route and reserving resources along the path if the call admission test is successful. The resource allocation depends on the division of the end-to-end QoS requirements into local QoS requirements at each link of the path from the source to the receiver. The call is admitted or rejected depending on whether the delay bound can be guaranteed, which in turn depends on the amount of resources available in the chosen route. The QoS guarantee depends on the scheduling algorithms used by the various nodes in the chosen path. A number of studies conducted in the literature [{5], show that many of the scheduling algorithms are able to provide guaranteed delay bound provided the burstiness of the incoming traf- c is bounded (for example, shaped by a leaky bucket). Recently, Stiladis [6] and Goyal [7,8] have provided a general framework to determine the end-to end delay bound for a network in which the servers on the path may not use the same scheduling algorithms. In this paper, we propose a distributed and ecient algorithm to divide the end-to-end delay requirements into local node resource requirements. A general overview of issues involved in the reservation establishment has been provided in Shenker [9]. An eort in the similar direction was made by the members of the Tenet Group in the past using a scheme based on a heuristic [10] and assumed NPEDF (Non preemptive earliest deadline rst) schedulers at all the routers in the path. In their approach, a tentative reservation is made at all the routers along the chosen path and best delay is calculated in the rst phase. The dierence between the desired delay and the best delay is dened as the slack in the path which is then used to deallocate resources at routers along the path depending on their local contribution to the total end-to-end delay calculation in the rst phase. Their approach does not make any attempt to optimize resource allocation and fur-

3 thermore it cannot be used with GPS-based scheduling disciplines since the delay bound calculation for these disciplines takes (delay) dependencies in the successive nodes into account and hence cannot be determined by simply adding the worst case delay contribution of all the routers in the path. Similar work has also been reported recently by Firoiu [11] for multicast connections, where the cell loss probability has been taken as the QoS parameter. They have proposed a proportional QoS division policy to divide the end-to-end QoS requirements into local QoS requirements. However, the division policy introduced in this paper is based on the cost function that takes the utilization of nodes into account. Here our goal is to demand less stringent QoS requirements from the highly utilized nodes in order to allow more calls to be admitted in the future. We give an example in which every node in the path implements PGPS scheduling algorithm. However, the approach used in this paper is a general one and can be also applied in the heterogeneous network environment, where dierent nodes use dierent scheduling schemes. The algorithm operates in three phases. In the rst phase, the schedulers operating at dierent nodes in the path send their respective latency parameters and spare capacity to the receiver. In the second phase, the nodes calculate their respective resource allocation using the distributed algorithm in order to meet the delay bound specied by the receiver. The algorithm developed in this paper may result in a delay bound that is tighter than that required by the receiver. The resulting slack produced is distributed to the respective nodes in proportion to their delay contribution. The algorithm is then further improved that results in the optimum allocation in the third phase. In Section we give a general expression of the end-to-end delay bound for the work-conserving GPS based scheduling schemes. The problem description and its formulation is introduced in Section 3. Section 4 introduces the QoS division policy based on the cost based function. Furthermore, it gives a distributed algorithm to implement it in a real system. Section 5 gives an improved algorithm to distribute the slack along the route in the third phase using Kuhn-Tucker conditions. Section 6 shows that the cost-based QoS division policy can be used for call admission control and resource allocation for multicast connection establishment. Section 7 gives a complete example based on the algorithm introduced in the Section 6. Section 8 gives the simulation results. The paper concludes with a discussion on our results along with the description of the work to be done in the future. End-to-End Delay Bound The delay bound on the packets of a session that conform to leaky bucket parameters (b; r) and is served by K servers can be given by [8]: 3

4 K?1 b X R b R j KX j ; (1) where, R j is the rate allocated to the session at router j in the path, R b is the minimum allocated rate and should be at least r in order to satisfy the stability of the queue, is the maximum packet length of the session under consideration and j is a constant that can be completely characterized by the scheduling algorithm used at node j and propagation delay between node j? 1 and j. The derivation of the above equation can be found in Stiladis and Varma [6] and Goyal and Vin [8]. Note the dierence between the above equation and the equation given in IETF [1] document and reproduced below, K?1 b X R b b R K X j ; () which depends only on the bottleneck rate in the path. This results in a conservative estimate of end-to-end delay even if dierent rates can be allocated at dierent servers in the path. For example, let us consider packets of a session with maximum length 100 bytes being served by two servers with allocated rates 3 kb/s and 64 kb/s respectively. Then, the middle term of () evaluates to 50 msec which is signicantly higher than the 37:5 msec given by the corresponding term in Equation (1). We observe that the end-to-end delay bound given in Equation (1) consists of three components: P K j: We have already stated that this term is entirely characterized by the scheduling algorithm and the propagation delay. The nodes in the chosen route can provide the information about this component through signalling messages. Hence it can be easily calculated once the route through the network is determined. P K?1 =R j:this term is the most important component since it is a function of the allocated rate at the servers which can be controlled through bandwidth allocation. b= b R: This is the only term in the delay bound equation that depends on the leaky bucket parameters. It has partial dependence on the allocated rates through the bottleneck rate b R r. If the bottleneck rate is same as the average rate r, then this component in the delay equation is completely determined by the leaky bucket parameters. 4

5 3 Problem Description and Formulation: We assume that path P that is most likely to meet the end-to-end delay requirements has been already determined by some routing algorithm. The problem is now to determine whether the chosen path P is able to meet the end-to-end delay bound requirements for the connection under consideration. The other problem that we address is the ecient allocation of resources at the nodes along the path to meet the delay bound. The problem can be stated as follows: Given a connection and the path P between the source S and destination D, based on the available resources (R j ) jp, determine whether the connection can be admitted on path P and if so, reserve the necessary resources. We make the following assumptions while deriving the algorithm: The path P from the source node S to the destination node D is already determined using some routing algorithm. The desired QoS is determined by the receiver. Sucient buering is provided at all the nodes to ensure that no packet loss occurs. We make the following additional comments regarding the last assumption. The burstiness of the connection increases as its travels through the network due to the deviation of the scheduler from the ideal uid-ow model. If every node implemented the uid-ow scheme perfectly then the required buering would simply be b (the token bucket size). The minimum amount of buering actually required (b reqd ) at a node j to ensure zero packet loss is [6,1]: b reqd = b min ij R i 4 j?1 X k=1 X j?1 R k k=1 k 3 5 : (3) Here, we have assumed that there is sucient buering at all nodes and hence there will be no packet loss at any node. However, if there is a scarcity of buers then the above equation can be used to calculate the buer requirements at dierent nodes. The j parameter for several well known work-conserving scheduling algorithms [8,7,6] are given in the Table 1. In the table, max;j is the maximum length of the packet served by the server at node j, V j is the maximum number of sessions that can be backlogged at the server and C j is the server capacity. We may rewrite the expression for end-to-end delay bound as, 5

6 Table 1 atency of several GPS-based schedulers Server j, atency Constant Virtual Clock (VC) Packet-by-Packet Generalized Processor Sharing Self Clocked Fair Queuing (SCFQ) d b min j R j K?1 X R j KX max;j C j max;j C j max;j C j (V j? 1) j : (4) Now since the last term can be determined by the scheduling algorithm parameters in advance, the delay budget to be satised by the intermediate nodes through proper resource allocation in the path can be given by, d m = d reqd? 0 K j 1 A ; (5) where d reqd is the end-to-end upper bound on delay, and d m is the upper bound on the remaining delay to be met by the nodes along the selected path through proper allocation of resources. et S = fs 1 ; S ; : : : ; ; : : : ; S K g be the available spare capacities at node 1; ; : : : ; j; : : : ; K respectively. The best delay bound is obtained when all the available resources at each hop are given to the connection under consideration and can be given by, d best = b min j K?1 X KX j : (6) If the desired delay bound for the application is greater or equal to d best then the call can be admitted, otherwise it must be rejected. The resource allocation problem can be formally stated as follows: Given the trac specications (b; r) of the connection and path P, nd a resource allocation, R = fr 1 ; R ; :::; R j :::R K?1 g, at nodes belonging to path P under the following constraints: h(r) 4 b min j R j K?1 X R j d m ; R j r 8j: (7) 6

7 4 End-to-End QoS Division Policies In this section we describe two policies for dividing the end-to-end QoS requirements into local QoS requirements. First policy tries to divide the QoS requirements evenly among the nodes, whereas the second policy tries to allocate it based on the utilization at the nodes. 4.1 Equal Division Policy In this scheme [1] each node allocates an equal amount of resources in order to meet the end-to-end delay bound. This scheme works as follows: The source node S sends the leaky bucket parameters (b; r) and the intermediate nodes along the path send the parameters associated with their respective scheduling algorithm j in the forward direction towards the receiver. The receiver node R based on its end-to-end delay requirements d calculates the rate to be allocated at the nodes along the route as follows: R equal = max " r; b (K? 1) d? P K j # : (8) This policy is very simple from the implementation point of view. Furthermore, it does not impose any signalling load on intermediate nodes since the receiver determines the rate for each intermediate node. 4. Division Policy Based on Cost Function The even division policy is not good for highly utilized nodes, since it would result in higher blocking for future call requests through them. Another approach would be to demand less stringent QoS from the highly utilized nodes. Here, we propose a new cost function based QoS division policy and dene the following cost function for a node j that takes the utilization of the node into account: C j =? R j : (9) We observe that as the allocated rate R j approaches the spare capacity, the cost function goes to innity. Resource allocation based on the minimization of this cost function would assign less stringent QoS requirements to highly utilized server nodes. et us dene node i as the bottleneck node if it has the 7

8 least spare capacity,i.e., S i ; 8j. If more than one node has the smallest spare capacity in the route then we designate one of them as the bottleneck node. We increase its contribution to the aggregate cost function by a factor to ensure that it gets the minimum allocated rate. The aggregate cost b function along the selected route can be given by, f(r) = K?1 X j6=i (? R j ) S i b S i? R i! : (10) We accordingly modify the constraints given in Equation (7) to K?1 b X R i R j d m ; R j r 8j: (11) Note that in the Equation (11), the rst term of the rst constraint depends upon R i, while that in Equation (7) depends on the node with the smallest R j. We will later show that the solution using this constraint also satises Equation (7). Now we develop the algorithm without taking the minimum bandwidth constraint, i.e. R j > r, into consideration. In the next section we will improve the algorithm by taking this constraint into account. We form the following agrangian for optimum resource allocations at the nodes along the selected path P : K?1 X j6=i b? R j S i under the following constraint, S i? R i K?1 X j6=i K?1 X j6=i 1 b R j R i A ; (1) b = d m : (13) R j R i Dierentiating Equation (1) with respect to R j obtain, and equating to zero we R j = S p j ; 8j: ( ) 1 p () (14) 8

9 From Equation (14) and constraint Equation (13), we obtain, h 1 q = d m? P i K?1 1 b S P i K?1 () p b : (15) p Si Eliminating between Equation (15) and Equation (14), we obtain the following expression for the required rate at node j, R j = P K?1 k=1 p Sk ( P K?1 k=1 p Sk p b Si ) b p Si q [d m? P K?1 k=1 :: (16) S k b S i ] The rate allocation at intermediate nodes can be done as follows: R cost = max [r; R j ] : (17) We can rewrite the above equation using Equations (6) and (5) as follows: 1 = 1 q 1 R j 1 d reqd? d best PK?1 p k=1 Sk p b Si A : (18) Here we observe that the above expression of R j is an increasing function of spare capacity. Hence with the use of above expression, the minimum rate would be allocated to the bottleneck node i. This shows that the results obtained under the modied constraints (11) conform with the constraints (7) originally stated in our problem denition. The resource allocation based on the above equation results in an improved allocation since it demands less resources from the highly utilized server nodes Algorithm The division policy based on the cost function can be implemented in either two or three phases in a distributed manner. The two phase implementation can be achieved by providing information about the spare capacities available at all the links in the path to the receiver. Here we propose a three phase implementation that avoids excessive advertisement and reduces the load at the receiver. It will require few additional elds in RSVP messages. We propose to include two elds IS and IS sqrt in the Adspec object of the Path message in order to calculate the sum of the inverse of spare capacity and sum of the inverse of square root of spare capacity respectively along the chosen path. We also include a eld to calculate the cumulative value of latency of the 9

10 service disciplines along the chosen route. The algorithm can be implemented as follows: Phase 1 The sender node initially sets the, IS and IS sqrt eld to zero and sends its leaky bucket parameters (b; r) and the maximum packet length of the requested connection in the Path message. A check is made at every intermediate node to determine if the spare capacity available is greater than or equal to the average rate of the incoming connection. If not, then it sends a ResvErr message towards the source. Intermediate node j updates the IS and IS sqrt eld as follows: IS? IS 1 ; IS sqrt? IS sqrt 1 q ; where is the spare capacity at node j. The bottleneck rate and the identity of the bottleneck are also determined in a distributed fashion and sent towards the receiver. Intermediate nodes also update the cumulative latency in the path as follows:? j ; where j is the latency factor associated with the scheduling algorithm used at node j. Phase : The receiving node R calculates the best possible delay bound d best from the information provided in the Path message using Equation (6). If the desired delay bound (d reqd ) for the application is greater or equal to best possible delay bound (d best ) then the connection is accepted else a ResvErr message is sent to the source node S. The receiving node now sends the accumulated value of IS, IS sqrt and the desired delay bound, d reqd, towards the sender S in the reverse direction. It also initializes the slack eld (Slack) to zero. Intermediate node calculates the allocated rate R j using the Equation (16). If the allocated rate based on the algorithm at a node j happens to be less than the average rate r then the average rate r is allocated at the node and the slack eld is updated as follows: Slack? Slack 1 R j? 1 r! : For the bottleneck node the multiplicative factor would be (b ) instead of in the second term of the above expression. The bottleneck rate along the selected path P is also sent towards the sender. 10

11 Phase3: The distributed nature of the algorithm may cause the resulting delay bound to be tighter than the desired delay bound d. The dierence is calculated as `slack' and is dened as follows: Slack = desired delay bound? delay bound obtained after the second phase The `slack' may be generated in the network, if the calculated value of the allocated rate at a node happens to be less than the average rate r. The value of cumulative sum of `slack' thus produced in the route is provided to the source node through the slack eld Slack by the intermediate nodes in the second phase. The sender node now sends the Slack value, if any, along the forward direction. The intermediate node can now deallocate resources in by using a portion of the slack. The care is again taken that the allocated rate at a node does not fall below the average rate r after deallocation. The slack distribution can be done as follows: Simple Scheme: The sender sends the `slack' value in the Slack eld of the slack redistribution message. Each node in the path grab a portion from the eld and decrement its value accordingly and forwards it to the next downstream node, if its value is greater than zero. To make sure that the allocated rate does not fall below the bottleneck rate calculated in the second phase, the bottleneck rate is also sent in the slack redistribution phase. This method is however unfair in slack distribution to the nodes far o from the source node. Proportional Division Scheme: In this approach, an additional eld is added in the second phase message to calculate the cumulative sum of the inverse of the allocated rates at the dierent nodes. There is no need for the nodes with allocated rates equal to the average rate r to participate in the process since they can not recover any slack. Each node j in the slack redistribution phase claims a portion of the slack (Slack j ) in proportion to its allocated rate as follows: Slack j = 1=R j Pj 1=R j Slack; 8j: (19) It must be ensured here that the allocated rate after deallocation of resources should not fall below the bottleneck rate calculated in the second phase. The amount of the slack used is subtracted from the `slack' eld before forwarding the redistribution message to the downstream node. This process continues till either the message reaches the receiving node or the slack eld becomes zero. It is to be noted here that it may not be possible to use whole slack due to the distributed nature of the algorithm. 11

12 5 An Improved Algorithm We note here that in the last section, we ignored the minimum allocated rate constraints, g j (R j ) = R j r 8j, while developing the algorithm. In this section we will modify the algorithm by taking these constraints into account through Kuhn-Tucker Conditions [13]. We will show at the end of this section that the new algorithm can be used to do the optimal allocation of resources in the third phase. et us dene I as the set of nodes along the given path for which the minimum allocated rate constraint is binding. Hence we have, g j (R j ) 4 R j = r; j I (0) g j (R j ) 4 R j > r; j 6 I: (1) Using where, and we obtain, 5f(R) 5 g(r) 5h(R) = : j : K?1 g ; 5 1 g(r) = fg 1 (R 1 ); g (R ); : : : ; g j (R j ); : : : ; g K?1 (R K?1 )g h(r) = 0 b K?1 X A ; R i R j j = (? R j )? R j j I; () R j = 1 S r j j 6 I: (3) For Kuhn-tucker condition to hold j > 0; 8j I. We show in the appendix that it always hold for the cost based resource allocation scheme used in this paper. Now using the delay constraint equation given in (13) we have, h 1 p = d b m? P P r ji r j6i P j6i i p 1 : (4) 1

13 Eliminating between Equations (3) and (4), we get the following expression for the rate to be allocated at node j: R j = 8 >< >: Pj6I P h p ps j6i j d m? p Sj b r PjI r P j6i i j 6 I; r; j I: (5) The Equation (5) gives the optimum allocation of rates at all the nodes along the chosen route to meet a given delay bound. We note that Equation (16) gives the allocated rates at the nodes when the minimum allocated rate constraints are not taken into consideration. This may result in the violation of minimum allocated rate at some nodes. We avoided this situation by allocating average rate r to those nodes in the second phase. In other words, we made the minimum allocated rate constraint binding on those nodes. Thus, we can identify the nodes that belong to the set I in the second phase and can now use the Equation (5) to allocate rates to the remaining nodes in the third phase. This will require slight modications in the algorithm given in the previous section without any change in the distributed nature of the algorithm. 6 Multicast Session A general framework for admission control and resource reservation for multicast sessions has been recently given by Firoiu [11]. In multicast sessions, slack or extra allocation may take place for a receiver due to the stringent quality of service requirements put by some other receivers on a common path segment. Firoiu [11] has exploited this feature of multicast sessions to develop ecient resource allocation algorithm. The algorithm is general in nature and can be used if the QoS division policy exhibits the uniformity property. An assignment is considered uniform if the QoS requirements put by one receiver is greater (or less) than those imposed by another receiver at all links on their common path segment. The most stringent QoS requirement is then met on the common path segment. The receiver demanding less QoS requirement on the the common path segment can use the slack thus produced to deallocate resources in the remaining links. The formal denition of the uniformity property as given in [11] is as follows: 13

14 Denition 1 (Firoiu, Towsley 1994) et P be a QoS division policy, P i = (S; D i ); i = 1; be two paths sharing a common path segment P 1 T P ; Q(S; D i ); i = 1; their end-to-end QoS requirements and (Q l i ) l(p i ) ; i = 1; the results of applying the policy P on P 1 and P. P is said to be uniform if: either Q l1 Q l 8l (P 1 T P ) or Q l1 Q l 8l (P 1 T P ). We will now show that the uniformity property also holds for cost function based division policy. For convenience, we reproduce the general assumption [11] required for the uniformity property to hold for any division policy: Paths P 1 and P are established simultaneously or close enough in time such that the network parameters used for QoS computation for both P 1 and P are the same. This assumption holds when the multicast session is established for all the receivers at the same time. Theorem 1 The cost function based division policy exhibits the uniformity property. PROOF. et us consider two paths P k = (S; D i ); k = 1; that share a common part. Now we rewrite the Equation (18) as follows: R j k = 1 1 p C h i ; k = 1; ; (6) Sj where, C k h = 6 4 d reqd? d best PK?1 p Sj b p Si k : (7) The C h k ; k = 1;, as we observe from Equation (7), is a function of the trac characteristics of the connections, QoS requirement of the receiver and the parameters of the selected route. This means that the C h k ; k = 1; is a constant at all the nodes along the route for a path P k = (S; D k ); k = 1;. Now since we assume that the paths are being established simultaneously, the parameter will also be same for any two connections along the common link. Hence we can say the following about the rate allocation along the common route: if C h 1 < C h then R j 1 > R j 8j (P 1 \ P ) 14

15 if C h 1 > C h or then R 1 j < R j 8j (P 1 \ P ) (8) and thus the uniformity property holds. 6.1 Static Multicast Problem The resource allocation for a static multicast session can be accomplished by assuming that a number of unicast sessions have to be established between the source node and all the destination nodes in the multicast session under consideration. The resource allocation at all the intermediate nodes can be accomplished using the cost-based QoS division policy. Once all the resource reservation requests for a link come from all the destination nodes sharing that link, the maximum requested bandwidth for that link is reserved. This will cause slack to be produced for all the destination nodes requesting less bandwidth from the link that it is sharing with other destination node requesting more bandwidth. For example, if a destination node D 1 requests R l 1 and the destination node D node requests R l from the common link l then if R l > Rl 1, it will result in a slack of (1=R l 1?1=R) l for the destination D 1 from this link. The slack produced will be (b )(1=R 1 1? 1=R) 1 if the common link happens to be the bottleneck link for the destination D 1. Furthermore, no slack will be produced from any intermediate link for the destination node D due to the uniformity property. The slack can now be used to deallocate resources at the leaf link going to the destination. The only restriction imposed, while deallocating resources at the leaf links, is that the new allocated rate does not fall below the bottleneck rate in the path if the leaf link is not the bottleneck link. Otherwise the delay bound violation may take place due to the heavy dependence of the delay bound equation on the bottleneck rate. Figure 1 illustrates an example in which receivers D 1, D and D 3 request delay bound of d1; d and d3 sec respectively, where d1 < d < d3: The rates at links 1; ; 3 and 4 are determined by the receiver D 1 that puts the more stringent delay bound requirements on these links. Similarly the rates at links 5 and 6 are determined by the receiver D. We assume that link 1 is the bottleneck link for all the destination nodes. et us denote the rate requested by receiver D i from link l by Ri. l Then the slack produced for receiver D3 from link 1 and due to dominating receiver D 1 will be (b)(1=r 1 3?1=R1)(1=R 1 3?1=R1). Furthermore, it will get slack equal to the (1=R 5 3? 1=R) 5 from link 5 due to the dominating receiver D. The slack can be used to reclaim the resource along the leaf link 7 leading to the receiver D3. Similarly the receiver D will get the slack from link 1 and, that it can use to reclaim the resource along the leaf link 6. 15

16 d1<d<d3 S D1 (d1) D (d) D3 (d3) rate determined by destination D1 rate determined by destination D rate determined by destination D3 Fig. 1. The multicast tree with end-to-end delay bound requirements 1 3 S D1 di 4 D d3 d3-di Fig.. The new receiver joining an existing multicast session 6. Dynamic Multicast Problem D3 In the dynamic version of the problem, individual receivers can join or leave the session at any time during the life time of the multicast session. We assume that a static multicast tree is already established and optimized for a given set of source and destinations. Furthermore, we assume that all the nodes in the optimized static multicast tree know the delay bounds for the packets coming from the source. When a new receiver wants to join the multicast tree, the underlying routing algorithm (for example [14]) nds the node in the existing tree that is closest to the receiver under consideration. Since the delay bound from the intermediate node already in the tree to the source is known, the residual delay bound required from the joining receiver to the closest node in the existing tree can easily be calculated. The resource allocation for the new path from the joining receiver to the closest node can now be done using the algorithm given in the previous section. Note that this procedure may result in the rejection of new requests even when sucient resources are available, since the worst case delay in this case is the sum of two worst case delay bounds and does not take interdependence of the delay in the two path segments (one from joining receiver to the node in the existing tree and the other in the existing tree to the source) into account. An example is illustrated through 16

17 Table Spare capacities at dierent nodes Node 1 Node Node 3 Node 4 Node 5 Spare Capacity before allocation (in Mb/s) Spare Capacity after Equal allocation (in Mb/s) Spare Capacity after Cost based allocation ( in Mb/s) Figure. The diagram shows a static multicast tree consisting of receivers D 1, D and source S. Receiver D 3 wants to join this multicast tree and has endto-end delay requirements of d3. The underlying routing algorithm identies the intermediate node 4 in the existing multicast tree that is closest to the joining receiver D 3. The intermediate node 4 informs the joining receiver that it has a delay bound equal to di for the packets coming from the source. This implies that the new connection can be accepted if the path segment from the intermediate node 4 can meet the maximum delay bound of d3? di. 7 A Complete Example In this section, we are going to illustrate through an example the improvement in bandwidth due to the use of cost based QoS division policy over the equal QoS division policy. We assume that the selected path from the source node S and destination node D consists of ve server nodes and each node uses the PGPS scheduling algorithm as the service discipline. We further assume that the maximum length of the packets served by dierent nodes is the same and is equal to the maximum length of the session under consideration and all the servers in the route have the same capacity C = 100 Mb/s. We select the following parameters for a session: = 1 Kbyte, b = 1 Kbyte and r = 1:5 Mb/s. The rst row of the Table shows the spare capacities at dierent nodes in the selected route. We assume that the desired end-to-end delay of the incoming call can be achieved by reserving Mb/s under equal allocation scheme. The spare capacities at dierent nodes after doing rate allocation using equal division policy and cost function based division policy are being illustrated in the second and third row of the table respectively (Also see Figure 3). We see that there is an improvement in the spare capacities at nodes that have insucient capacities with the use of cost based QoS division policy. Here we would like to point out that though equal allocation policy is more 17

18 3.5 Spare Capacity versus Nodes Equal Allocation Policy Cost Based Allocation Policy 3.5 Spare Capacity ( in Mb/s) Nodes Fig. 3. Spare capacity versus nodes ecient in terms of overall bandwidth utilization, it is very poor in terms of bandwidth distribution along the route. Hence the use of cost based division policy proposed here, would cause an ecient distribution of spare capacities along the route, resulting in less blocking for future calls along the route. We conrm this latter through extensive simulations. Our nal example shows the resource allocation problem for a multicast session when the cost-based QoS allocation scheme is being used. Figure 4 shows an example case, where a multicast session is to be established between the source S and destinations D1 and D. The spare capacities available at dierent links are indicated near every link in the diagram. The end to end delay requirements of the destinations are also being indicated in the diagram. The diagram at the top of Figure 5 gives the requested rates at the shared links by both destinations. The bottom diagram in the gure indicates the allocated rates at dierent nodes after the optimization. We observe that the uniformity property holds along the common path and the destination node D1 is able to reduce its bandwidth requirements by 10:76 Mb/s along the leaf link. 18

19 D msec 40 S msec Fig. 4. The multicast tree with end-to-end delay bound requirements D 8.09 D1 S D a)the result of cost-based QoS division policy D S D b)the resource allocation after the slack recovery and optimization Fig. 5. The resource allocation with cost-based division policy, both before and after optimization 8 Simulation Results For the purpose of conducting experiments, random networks were generated by using the algorithm given in [14]. The distance between each pair is chosen from a uniform distribution between 0 and a maximum number (say D). An edge is introduced between nodes u and v with the following probability: 19

20 ?d(u; v) P (fu; vg) = exp D where d(u; v) is the distance between two nodes u and v. The parameter determines the edge density of the generated network and determines the ratio of shorter edges to the longer ones. We assumed that both these parameters are equal to 0:4 in our simulations. The parameters of the incoming sessions were assumed to be as follows:b = 1 Kbyte, r = 10 Mb/s, = 1 Kbyte. All the links in the generated networks were assigned 00 Mb/s capacity. The size of the network was limited to 50 nodes. The end to end queueing delay was assumed to be msec for all source destination pairs. The calls were generated according to a poisson process with parameter and their durations were assumed to be distributed exponentially with mean 1=. The parameter = = characterizes the load oered to the network, i.e., the average number of calls that would exist at any time in an innite resource network. For every incoming call, source node and destination node were selected randomly with equal probability from the nodes in the network and a path was selected using minimum hop routing algorithm. The call was rejected either due to inadequate bottleneck bandwidth (i.e., less than r) or delay-bound violation. Resource allocation was done for every successful call. Two sets of experiments were conducted by using the equal and cost based resource allocation algorithms with the same series of calls. For every experiment, 10 6 calls were generated. Figure 6 shows the call blocking probability under both equal and cost-based resource allocation schemes for a typical network. We observe that there is a signicant improvement in the call blocking rate with the use of cost based allocation scheme over equal allocation scheme. The results are within 95% condence interval and generated using method of independent replications. The results were veried over a number of random networks. We conducted another set of experiments to see the eect of ratio of maximum burst size b to the maximum packet length of the call by increasing the maximum burst size to Kbyte without changing the maximum packet length. The simulation results given in Figure 7 show that the equal allocation policy starts giving a better performance when the network is heavily loaded. This is due to the fact that as the ratio of b and increases, the bottleneck link becomes the major contributor to the queueing delay. Even a small decrease in allocated rate at a bottleneck link translates into very large resource requirement at remaining links in the chosen path. If this ratio is very large then equal allocation policy will outperform the cost based allocation policy. In a practical system, the maximum burst size is application dependent and the maximum packet length is determined by the underlying network technology. For example, in Ethernet the maximum packet size is limited to 1500 bytes. Hence the equal allocation or cost based allocation policy should be 0

21 0.1 short_hp_equal short_hp_cost 0.01 Blocking Probability e Offered load Fig. 6. The blocking probability vs. oered load:b= 0.1 short_hp_equal short_hp_cost 0.01 Blocking Probability Offered load Fig. 7. The blocking probability vs. oered load:b= 1

22 used depending on the application and underlying network technology. 9 Conclusions and Future Work In this paper, we have developed a solution to the problems of call admission control and resource allocation at nodes once the path is selected from the source node to the destination node using some suitable routing algorithm. In particular, we studied the call admission control and resource allocation problem for Guaranteed QoS service applications which require rm end-to-end delay bound. The algorithm that we have developed is distributed in nature and can be implemented in three phases. At the end of rst phase, the receiver determines whether the call can be admitted along the chosen route. If so, then the resources are allocated at intermediate nodes using the distributed algorithm. A tighter delay bound may result since we do not take the minimum allocated rate constraint while developing our algorithm. The resulting `slack' can be reclaimed in the third phase using a simple algorithm. We then improved the algorithm by taking the minimum allocated rate constraint into consideration using Kuhn-Tucker condition that resulted in an optimum allocation of resources. We have taken a simple example in which all the nodes in the route implement PGPS scheduling algorithm. The example shows that the use of cost based QoS division policy results in an improved distribution of the QoS across the intermediate nodes. We conrm this through extensive simulations. However, the relative performance is dependent on the application and the underlying network technology. We then extended our work to multicast sessions. In the future, we would like to develop suitable multicast routing algorithms that use the cost function introduced in this paper to nd the least cost multicast routes for guaranteed quality of services. Another interesting area of study would be to come out with new cost functions that are more ecient in distributing the QoS among the nodes along the chosen route. There is also a scope of improvement in doing the resource allocation for both static and dynamic multicast connections. A Appendix Here we show that the agrange multiplier j > 0; 8j I, i.e., the Kuhn- Tucker optimality condition is always satised for the cost based QoS division policy. We rewrite the Equations () and (3) as follows:

23 j = (? r)? r j I; = R i S i (S i? R i ) i 6 I: (A.1) We note that all nodes j that are included in the set I are those for which the minimum bandwidth constraint was violated in the rst phase. Furthermore, our algorithm allocates rates according to the amount of spare capacity available. Hence, we can say that S i > and R i > r i 6 I; j I: Now eliminating from Equations (A.1), we have j = (? r)? S i Ri r (S i? R i ) : Now, j > 0, implies that ) (?r) > Ri r S i (S i?r i ) ) (?r) > S i (S i?r i ) [since R i > r] ) 1 (?r) > 1 (S i?r i [since ) S i > ] ) (S i? R i ) > (? r) : We rewrite the Equation (6) as follows: R j = 1 q C h : We have,? R j = q C h 1 q C h : Now, since S i > ; j I; i 6 I, we have, (S i? R i ) > (? r) 8i; j. Hence the result follows. 3

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