The influence of QoS routing on the achievable capacity in TDMA based Ad hoc wireless networks

Transcription

1 Wireless Netw () 6:9 DOI.7/s The influene of QoS routing on the ahievable apaity in TDMA based Ad ho wireless networks S. Sriram Æ T. Bheemarjuna Reddy Æ C. Siva Ram Murthy Published online: August 8 Ó Springer Siene+Business Media, LLC 8 Abstrat The issue of providing Quality of Servie (QoS) guarantees in an Ad ho wireless network is a very hallenging problem. In this paper, we make the following ontributions: (i) analytially derive bounds for the end-toend all aeptane rate using existing queueing theory methods, (ii) study the impat of the routing sheme on the end-to-end all aeptane rate, and (iii) propose a differentiated servies sheme for deterministially providing QoS guarantees. Unlike the existing studies whih analyze the transport apaity, we fous on the end-to-end all aeptane. The framework that we assume is that of a TDMA based Ad ho wireless network. The routing sheme employed influenes the end-to-end all aeptane of the network. The metris that we onsider are the all aeptane probability and the system saturation probability (i.e., the probability that the network is in a state in whih every new all is rejeted). We derive general bounds on the all aeptane and the system saturation for the ase of differentiated-lasses of users in the network. These bounds indiate the number of alls of the highest priority lass that an be admitted into the network. Simulation studies were arried out to study the effet of S. Sriram Department of Computer Siene, University of California, Berkeley, CA 97, USA sriram_s@s.berkeley.edu T. Bheemarjuna Reddy (&) CalIT, Jaobs Shool of Engineering, University of California, San Diego, CA 99, USA btamma@usd.edu; arjun@s.iitm.ernet.in C. Siva Ram Murthy Department of Computer Siene and Engineering, Indian Institute of Tehnology Madras, Chennai 6 6, India murthy@iitm.a.in load, hopount, and the influene of the routing protool on the all aeptane. The inrease in the all aeptane rate with the introdution of load-balaning highlights the importane of load-balaning in enhaning the system performane. From these studies, we arrive at the following results: (i) load-balaning leads to signifiant improvement in the end-to-end all aeptane rate, and is an important fator in attaining the maximum end-to-end all aeptane rate in a given network and (ii) it is indeed possible to provide deterministi QoS guarantees for a designated set of nodes whih are haraterized by deterministi guarantee limit. Keywords Ad ho wireless networks QoS routing TDMA Call aeptane probability Load-balaning Introdution An Ad ho wireless network is a olletion of mobile nodes that an ommuniate over radio without any pre-existing infrastruture. Two nodes an ommuniate diretly with eah other if eah lies in the transmission range of the other. Two nodes that annot diretly ommuniate an do so in a multi-hop manner in whih the other nodes funtion as routers. Suh networks are used in military installations and in emergeny situations as they permit the establishment of a ommuniation network at very short notie. However, these networks are limited by onstraints in their bandwidth and power onsumption. For their widespread deployment, Ad ho wireless networks now need to support appliations that generate realtime traffi suh as voie and video. Suh traffi requires the network to provide guarantees on the QoS of the onnetion. The important aspets in the proess of providing

2 9 Wireless Netw () 6:9 suh guarantees are the routing protools that establish paths that an satisfy the QoS requirements and the reservation mehanisms that reserve the neessary resoures along the path. A problem of onsiderable interest in this regard is that of theoretially estimating the nature of the guarantees that an be provided by a QoS sheme. These estimates on the parameters of QoS routing protools give us an idea of the maximum guarantees that an be provided, and allow us to gauge how far the existing shemes are from the ideal limit. In this work, we onsider the problem of QoS routing for multimedia traffi (i.e., UDP traffi) in a TDMA based Ad ho wireless network, where the QoS onstraint on the alls is that of bandwidth. Our fous is the end-to-end all aeptane rate whih is a measure of the number of voie or video alls that an be admitted into the network. The alls arriving in the network belong to different lasses based on whih the requirements of the alls are prioritized. Thus, the parameters that we fous on are: the all aeptane probability and the system saturation probability. The variation of these parameters enables us to answer questions suh as () What are the maximum number of high-priority alls that an be sustained in the network at a given load?, () What is the likelihood that the network enters a state where no more new alls an be aepted?, () What is the effet of the routing protool on the all aeptane?, () How lose to the theoretial limit do the routing protools approah? Then, we address the problem of ensuring deterministi all aeptane for a ertain sub-set of the alls. We estimate the deterministi guarantee limit whih is a mobility-independent measure of the number of high-priority alls that an be admitted into the network. We also determine the all aeptane probability for the lasses for whih deterministi guarantee annot be provided. In this work, we model the network at the level of transmission range of eah node. The range of a node is analyzed as a Markov proess where the alls are the entities to be servied. The reservation of slots for the all onstitutes the servie of the all. The modeling of a wireless network as a olletion of Markov proesses is unique in that, due to the loal broadast nature of the hannel, the reservation of slots in the transmission range of a node affets the status of the slots in the neighboring regions. Capturing this property of wireless networks is essential to model the harateristis of the network aurately. Suh a modeling must also be able to reflet the harateristis of the routing protool used. We begin by analyzing a general ase of a network that an support multiple-lasses of alls where preemption of alls does not exist. We then provide a losed-form estimate of the all aeptane probability and the saturation probability for the ase of a single-lass of users and disuss the probabilities for the highest-priority lass in the preemptive ase. We ompare the all aeptane probabilities of shortestpath routing and two routing protools that attempt loadbalaning. Finally, we estimate the deterministi guarantee limit. The rest of this paper is organized as follows: Set. briefs the related work in this area, Set. presents the system model and derives theoretial bounds, Set. presents load-balaning shemes, Set. disusses the details of the simulation, and Set. 6 presents the simulation results. Finally, Set. 7 onludes the paper with diretions for future work. Related work In their seminal work [], Gupta and Kumar introdued a random network model for studying throughput saling in a fixed wireless network. It was shown that even under optimal onditions, the transport apaity (bit-distane produt that an be transmitted over the network) of the pffiffi network is hð n Þ bit-m/s, where n is the number of nodes present in the network, for the protool model onsidered. Further they showed that in suh a random network the throughput sales as h pffiffiffiffiffiffiffiffi per soure-destination pair. nlogn In [], the authors studied saling laws for the transport apaity as the value of n inreases. Further, the optimality of multi-hop operation is provided in some situations. In [], the authors showed that by allowing nodes to move, the throughput saling hanges dramatially. They showed that if node motion is independent aross nodes and has a uniform stationary distribution, a onstant throughput saling (h()) per soure-destination pair is feasible. In [], the authors obtained lower bounds on the apaity of ad ho networks with two types of non-uniform traffi patterns. In [], the authors onsidered power onstrained ad ho networks and demonstrated that throughput apaity inreases with node density, in ontrast to previously published results. This is the result of the large bandwidth, and the assumed power and rate adaptation, whih alleviate interferene. In [6, 7], the authors analyzed the performane of IEEE 8. DCF based single-hop wireless networks. In [8], the authors proposed a methodology to analytially ompute the throughput apaity, or the maximum end-to-end throughput of a given soure and destination pair in an IEEE 8. DCF based multi-hop wireless network. They onsidered two key fators whih affet the end-to-end throughput apaity: (a) neighboring ontentions and (b) hidden node interferene. While previous studies analyze the transport apaity of ad ho networks, in this work our fous is on the end-toend all aeptane rate whih is a measure of the number

3 Wireless Netw () 6:9 9 of alls with end-to-end bandwidth reservation that an be supported by the network. The previous studies on transport apaity study how it sales with the number of nodes. We study the dependene of end-to-end all aeptane rate on the network load and the routing protool. The framework that we assume is that of a TDMA based network. The routing sheme employed influenes the end-toend all aeptane rate of the network. In this paper, we investigate the end-to-end all aeptane rate and the influene of shortest-path routing and load-balaned routing protools on it. Theoretial analysis We onsider an Ad ho wireless network omprising N nodes uniformly distributed at random in a terrain of area A. The transmission range of eah node is R. We assume the presene of a slotted TDMA mehanism at the MAC layer. The hannel time is divided into super-frames whih in turn are divided into time slots for the transmission and reeption of pakets by nodes in the network. Time synhronization requires keeping aside some fration of available time slots in eah super-frame to ahieve synhronization among nodes in the network. Numerous solutions are proposed in the literature to address time synhronization issue in TDMA based wireless networks [9 ]. One ould employ any of those existing solutions to ahieve time synhronization. Eah super-frame onsists of B time slots after exluding those time slots that are kept aside for time synhronization. A node has to reserve one or more slots for ommuniating with its neighbors. It is also possible to reuse the slots spatially depending on the interferene pattern of the nodes. This is the key idea that is used in deriving the bounds. We define all as a voie or video session onsisting of many pakets. A all is said to have been set up between two nodes i, j if there is a set of nodes (p = i, p,, p n = j) suh that p k? is in the transmission range of p k (p k? [ region R(p k )) and there is a permissible shedule for transmission of pakets from node i at eah node p k ( B k B n - ). In the absene of preemption, finding suh a shedule is equivalent to finding a set of free slots in eah region R(p k ). The definition of a free slot in a region omes later in Set. B. The bandwidth of a all is measured in terms of the number of slots used for transmission. A all is setup by reserving slots along the path of the all. A node may either transmit or reeive in a partiular slot (a node is said to reeive in a partiular slot if any of its neighbors is transmitting in that slot). A slot is said to be free at a node j, ( B j B N) if it is neither transmitting nor reeiving during that slot. For a node j to transmit in a partiular slot, the slot must be free at node j and none of the nodes lying in its transmission range must be reeiving in that slot. For a node j to reeive in a partiular slot, the slot must be free at node j. This definition permits node 6 to transmit to node in the same slot as the one used by node to transmit to node in Fig., provided nodes and do not hear from node 6. On the other hand, in the sender s range, node must use a different slot to transmit to node beause node hears the transmission by node. (A) System model: Consider a network NW = {,, N} of N nodes that an support K lasses of alls where lass i alls have a higher priority than lass j ( B i \ j B K) alls. We would like an estimate of how many alls of a partiular lass an be supported. This implies that we an definitely support suh a number of lass alls where lass is the highest priority lass. If all the available slots are oupied by alls of various lasses upon arrival of a lass j all, one or more alls of lower priority lasses an be preempted based on the bandwidth requirement of lass j all that has arrived to ensure that the arrived all be aepted. Thus, we would like to provide a guarantee on the number of alls of a partiular lass that an be aepted. Assumptions made for the analysis are the following: Calls of a partiular lass-k arrive at eah node distributed aording to a Poisson proess of mean k k. We assume that the alls of all lasses have equal bandwidth requirements: eah all requires reservation of a single slot in the super-frame. The reserved slot is being used for transmitting one paket of the all in eah super-frame till the duration of the all ends (i.e., all departs from the network). The duration of a all (voie or video session) is exponentially distributed with mean duration l k : We do not take node mobility into aount in the estimation of all aeptane and system saturation. Fig. An example of possible transmissions 6

4 9 Wireless Netw () 6:9 (However, the deterministi guarantee limit is independent of mobility.) We assume that the routing algorithm is suh that for any path found by the algorithm, the number of nodes on the path that lie within the transmission range of any node on the path (inlusive of the node itself) is not greater than some onstant. In the absene of suh an assumption, it is possible to onstrut a senario (Fig. ) where a single all needs to use all the slots in the system. In Fig., eah of the nodes on the path is in the transmission range of the other nodes. So the use of a slot for transmission by one of the nodes implies that the slot annot be re-used by the other nodes on the path. Thus, if node A transmits to node B on slot #, slot # annot be used by any of the other nodes to transmit to their downstream nodes. If we were to onsider a P-hop path with the nodes in the onfiguration given in Fig., the number of slots used would be P. Hene, it would be diffiult to provide a bound on the number of alls that an be admitted. This property is satisfied with = for protools that ensure that if a path is to be set up from A to C, the path used is the link (A, C) rather than links (A, B) and (B, C), where A, B, and C are nodes suh that eah an listen to the other two. This an be done by using an appropriate forwarding of the route request pakets in whih a node drops all exept the first route request that it reeives. We assume that interferene range and the transmission range to be the same for all nodes in the network. We denote the interferene range by Q and the transmission range by R. When we onsider the more general ase of Q greater than R, the parameter will be higher than in the ase where Q = R. So hannel reuse redues further. Ratio of Q R ditates the number of slots needed for a multi-hop all. As Q R inreases more number of slots are needed for multi-hop alls. We study this ase in Set. B.7. (B) Analytial bounds: Initially, we assume that all preemption does not our. We derive upper and lower bounds on the all aeptane probability for the ase of single-hop and multi-hop alls respetively. Consider a node j and the region spanned by its transmission range R(j). Any all passing through R(j) uses up some number of slots. The number of slots used up in the region R(j) depends on the number of alls originated from node j, the number of alls from any of the neighbors of node j, the number of alls that originate from outside R(j) and are terminated at some node in R(j), and the number of alls that originate from outside R(j) and are routed through R(j). A slot is said to be free in R(j) if no nodes in R(j) are either transmitting or reeiving in that slot (i.e., slot is free at all nodes in region R(j)). In Fig., nodea transmits to node B on slot. Node D transmits to node B on slot. Node E transmits to node C on slot. If the network had a total of slots, the free slots at node A would be {,,,} while the free slots in the region R(A) would be {,}. We an thus view R(j) as a server of slots for whih the alls ontend. Although the distribution of all arrivals of a partiular lass at eah node is known to be Poisson, the distribution of alls arriving at R(j) is not Poisson due to the splitting of the Poisson streams (Consider alls arriving at a node based on a Poisson proess of mean k. Assume that the node has to forward the all along one of two links. If the node forwards alls in a non-random manner, the arrival of alls at the downstream node will no longer be Poisson). We make use of Kleinrok s Independene Assumption, aording to whih, for moderately heavy all arrival at eah node, the net all arrival at the region R(j) A # # B C # D R E # # F D # B # A # C E Fig. An example senario #x slot number x Fig. An example to distinguish free slots at a node and free slots in a region

5 Wireless Netw () 6:9 9 an be regarded as Poisson. Thus, alls of a lass-k arrive at R(j) aording to a Poisson distribution with mean: k k ðjþ ¼ i¼n f k ði; jþk k i¼ where f k (i, j) is the fration of lass-k alls originating in node i that pass through the region R(j). This an be rewritten as: k k ðjþ ¼@ f k ði; jþþjnðjþj þ Ak k ðþ i6rðjþ where N(j) denotes the set of nodes in the transmission range of node j. The parameter f k (i, j) isdependent on the routing protool. For a protool suh as shortest-path routing, whih leads to heavy loads in the enter of the network, f k (i, j) would be high for nodes j ( B j B N) loated near the enter. For protools that implement loadbalaning, the value of f k (i, j) should be fairly uniform aross the nodes. The state of the system R(j) is given by the number of alls of eah lass being served (eah of whih uses up some of the slots of R(j)) by R(j). We thus model R(j)asaK-dimensional disrete-time Markov proess (t) = (n,, n K ), where n k denotes the number of lass-k alls being served by R(j) at time t []. The use of a Markov proess is appropriate here beause for these wireless networks, the slot alloations at any instant of time only depend on the alloated slots at the previous instant and the arrivals and departures of alls in that instant. A proess that has greater dependene on the past may be appropriate in modeling systems whih attempt to ahieve long-term fairness of slot alloation, for example. However, for the setting onsidered in the paper, a first-order Markov proess is a natural model. We denote: P((n,, n K ) (n,, n K )) = P((t? Dt) = (n,, n K ) (t) = (n,, n K )) as the probability that the system R(j)isinthestate(n,, n K ) at time t? Dt given it is in the state (n,, n K ) at time t. Pððn ;...; n k þ ;...; n K Þjðn ;...; n k ;...; n K ÞÞ ¼ k k ðjþdt ðþ Pððn ;...; n k ;...; n K Þjðn ;...; n k ;...; n K ÞÞ ¼ n k l k Dt; n k [ ðþ We have used a disrete-time Markov hain in our setting. The approximation used in this setting however is In the most general ase of a model orresponding to K lasses of KþB alls in a network having B slots, the Markov proess has B states. This is not a problem for the urrent analysis sine the transitions between the states are restrited: every state has at most K neighboring states, and the proesses assoiated with any given regions are deoupled. Further, we are interested in only the steady state of the proess and not in the paths traversed. The state-explosion needs to be takled for an analysis that onsiders oupled proesses or preemptive alls: the interested reader may refer [] and []. that the probability of more than one all arriving or departing within a super-frame is low. This is a valid approximation when the super-frame lengths are small (as is the ase for the ase for wireless networks with high bandwidth). In this senario, the system has transition probabilities that are similar to those for a ontinuous-time Markov hain for a small interval. In effet, the system has transition probabilities that are idential to those desribed in above equations. The state-transition diagram representing the transitions into and out of one of the states of the Markov proess is shown in Fig.. The Markov proess has a unique steady-state probability distribution []. Using Eqs. and along with the normalization of probabilities, we an alulate the probability that the system is in a partiular state (n,, n K ) as: Pððn ;...; n K ÞÞ ¼ Yk¼K q k ðjþ n k ðþ GðjÞ n k! k¼ Q k¼k q k ðjþ n k k¼ n k! where q k ðjþ ¼ k kðjþ l and GðjÞ ¼ P k n þþn K B is a normalization fator. We would now like to extend this Markov proess to distinguish between alls that terminate in a node in R(j) (all them type-u alls) and those that do not (type-v alls). Let us say that a fration f of the alls terminate in some node in R(j). If the destination were to be hosen randomly, then f ¼ jnðjþjþ N : The state of the system is now given by: ðn ;U ; n ;V ; n ;U ; n ;V ;...; n K;U ; n K;V Þ where n k,u is the number of lass-k alls that are type-u alls in R(j) and n k,v is the number of lass-k alls that are type-v alls. The probability that the system is in a state (n,u, n,v,, n K,U, n K,V ) is: Pððn ;U ; n ;V ;...; n K;U ; n K;V ÞÞ ¼ Yk¼K q k;u ðjþ n k;u q k;v ðjþ n k;v EðjÞ n k;u! n k;v! where k¼ q k;u ðjþ ¼ f k kðjþ ; q k;v ðjþ ¼ ð f Þk kðjþ l, and EðjÞ ¼ k l Q k k¼k Pn ;U ;n ;V ;...;n K;U ;n K;V k¼ q k;u ðjþ n k;u n k;u! q k;v ðjþ n k;v n k;v! is a normalization fator. The probability that the system is in a state (n,v, n,v,, n K,V ) (a state in whih there are n,v lass- type-v alls, n,v lass- type-v alls, and so on) is: Pððn ;V ;...; n K;V ÞÞ ¼ Yk¼K q k;v ðjþ n k;v ðþ HðjÞ n k;v! k¼ For the ase of preemption, the system an move between ertain other states. Corresponding to the ase of preemption of a lass- all by a lass- all, the system an move from the state (n, n,, n K )to (n?, n -,, n K ), n C.

6 96 Wireless Netw () 6:9 Fig. The transitions into and out of one of the states of the Markov proess representing the region R(j). For the state (n, n,, n K ), n [, n [,, n K [ (n, n,..., n,..., n ) i (n +, n,..., n,..., n ) i K (n, n +,..., n,..., n ) i K K n µ t λ ( j) t (n +)µ t λ ( j) t (n +) µ t λ ( j) t K λ ( j) t (n, n,..., n,..., n ) i K n µ K t K (n, n,..., n,..., n ) i K µ t n λ ( j) t K λ ( j) t (n +)µ i t i λ ( λ ( j) t i i n i (n +) µ K K t (n, n,..., n +,..., n ) i K j) t µ i t (n, n,..., n,..., n ) i K (n, n,..., n,..., n ) i K (n, n,..., n,..., n +) i K where HðjÞ ¼ P Q k¼k q k;v ðjþ n k;v n ;V þþn K;V B k¼ n k;v! is a normalization fator. () Call aeptane probability: In this setion, we are going to derive the all aeptane probability of both single-hop and multi-hop ases for a non-preemptive system (a system where the aepted alls are not dropped for a new all). Lemma P(Number of used slots in a region R(j) B x) B P(Number of type-v alls in R(j) B x), where x N: Proof For every type-v all, at least one unique (till then unused) free slot in the region R(j) must be used (see Fig. ). Thus: Theoretial upper bound for probability of all aeptane: We now derive an upper bound on the probability of all aeptane for the ases of single-hop and multi-hop alls. Single-hop ase: Consider a single-hop all from node j to its neighbor node l. For the all to be aepted, at least one slot must be free in the region R(j). Thus P A (probability of a single-hop all is aepted) is: P A ¼ PðNumber of free slots Þ ¼ PðNumber of used slots B Þ where B is the total number of slots in the system. From Lemma : Number of type-v alls in RðjÞ [ x ) Number of used slots in RðjÞ [ x and Number of used slots in RðjÞx ) Number of type-v alls in RðjÞx Hene PðNumber of used slots in RðjÞxÞ PðNumber of type-v alls in RðjÞxÞ h Lemma P(Number of alls in a region R(j) B x) B P(Number of used slots in a region R(j) B x), where x N and is the routing-algorithm dependent onstant fator that denotes the maximum number of nodes on a path that lie within the transmission range of any node on the path. Proof P A PðNumber of type-v alls B Þ PðNumber of type-v alls [ B Þ PðNumber of type-v alls ¼ BÞ P A n ;V þn ;V þ...þn K;V ¼B HðjÞ Yk¼K k¼ q k;v ðjþ n k;v n k;v! ð6þ Number of alls in RðjÞx ) Number of used slots in RðjÞx Hene PðNumber of alls in RðjÞxÞ PðNumber of used slots in RðjÞxÞ h

7 Wireless Netw () 6:9 97 C C j C P P C P P P C Slot # Slot # Slot # Fig. In the region R(j), C and C are type-u alls; C, C, and C are type-v alls. For eah type-v all, we see that at least one slot that has not been used so far in R(j) must be used. For the type-u alls, slot reuse is possible in some ases For the ase of a single-lass of alls, Eq. 6 redues to P A q ;V ðjþ B ð7þ HðjÞ B! Multi-hop ase: We set the onstant =. Consider a (M - )-hop all (M C ) setup along the nodes (p,, p M ). When a slot is reserved for transmission between p and p, the total number of free slots at R(p ) dereases by (sine the slot annot be used for transmission from p to p ). Thus, the total number of slots available at R(p ) an be onsidered Fig. 6 Multi-hop all setup. R(P ) needs slot # to be free. R(P ) now annot use slot # and requires slot # (some other slot) to be free. R(P ) annot use slots # and #, and requires slot # (any other slot) to be free. R(P ) an transmit in slot # if it is free A all is suessfully forwarded in region R(j) if slots an be found in R(j) so that the all having arrived at node j is forwarded to its next hop in the path. For the all to be aepted, it must first be suessfully forwarded in the region R(p ), must then be suessfully forwarded through eah of the regions R (p ), R (p ),, R (p M- ). A neessary and suffiient ondition for suessful forwarding is the presene of at least one free slot in eah of the intermediate regions. Thus P A is given by: P A ¼ PðSuessful forwarding of all in Rðp ÞÞ PðSuessful forwarding of all in R ðp ÞjSuessful forwarding of all in Rðp ÞÞ PðSuessful forwarding of all in R ðp ÞjSuessful forwarding of all in R ðp ÞÞ... PðSuessful forwarding of all in R ðp M ÞjSuessful forwarding of all in R ðp M ÞÞ P A ¼ PðNo. of free slots in Rðp ÞÞ PðNo. of free slots in R ðp ÞjSuessful forwarding of all in Rðp ÞÞ PðNo. of free slots in R ðp ÞjSuessful forwarding of all in R ðp ÞÞ... PðNo. of free slots in R ðp M ÞjSuessful forwarding of all in R ðp M ÞÞ ð8þ as B -. Call this modified region R (p ). When slots have been reserved between p and p,andbetweenp and p, the number of free slots at R(p ) dereases by so that the total number of slots at R(p ) an be regarded as B -. Call this modified region R (p ). The number of slots, for the regions R(p ),, R(p (M-) ), is thus effetively, B - (sine = ). (Thus, aording to this notation, a region R (j) hasonefewerslot,whiler (j) hastwo fewer slots). A multi-hop all setup for M = is shown in Fig. 6. From Lemma : P A PðNo. of type-v alls in Rðp ÞB Þ PðNo. of type-v alls in R ðp ÞB Þ PðNo. of type-v alls in R ðp ÞB Þ... PðNo. of type-v alls in R ðp M ÞB Þ

8 98 Wireless Netw () 6:9 P A... P Q k¼k q k;v ðp Þ n k;v Hðp Þ n k;v! n ;V þn ;V þ...þn K;V ¼B k¼ P Q k¼k q k;v ðp Þ n k;v H ðp Þ n k;v! n ;V þn ;V þ...þn K;V ¼B k¼ P Q k¼k q k;v ðp Þ n k;v H ðp Þ n k;v! n ;V þn ;V þ...þn K;V ¼B k¼ P Q k¼k q k;v ðp M Þ n k;v H ðp M Þ n k;v! n ;V þn ;V þ...þn K;V ¼B k¼ P Q k¼k q k;v ðp M Þ n k;v H ðp M Þ n k;v! n ;V þn ;V þ...þn K;V ¼B k¼ Q k¼k q k;v ðjþ n k;v n k;v!!!!!! ð9þ where H ðjþ¼ P n ;V þþn K;V B k¼ and H ðjþ¼ P Q k¼k q k;v ðjþ n k;v n ;V þþn K;V B k¼ n k;v! : For the ase of a single-lass of alls, Eq. 9 redues to " # P A q ;V ðp Þ B Hðp Þ B! " # q ;V ðp Þ B H ðp Þ ðb Þ! " # q ;V ðp Þ B H ðp Þ ðb Þ! " # q ;V ðp M Þ B... H ðp M Þ ðb Þ! " # q ;V ðp M Þ B H ðþ ðp M Þ ðb Þ! The RHS (Right Hand Side) of Eqs. 7 and are hard to solve for in a losed-form. For moderate-to-heavy traffi, q [ and the inequality remains valid if we replae q,v (p j ), B j B M - byq Max,V (the maximum value of q,v (p j ) aross all the regions). Denoting the RHS as P Max A : P Max q ;V Max B A ¼ for single-hop alls ðþ H B! " # " # P Max A ¼ Max q B ;V Max q B ;V H B! H ðb Þ! " # Max q B M ;V H for multi-hop alls ðþ ðb Þ! where H ¼ P b¼b b¼ H ¼ P b¼b Max q b ;V b¼ q ;V Max b b! : b! ; H ¼ P b¼b b¼ q ;V Max b b! ; and Theoretial lower bound for probability of all aeptane: In this setion, we derive lower bounds on the probability of all aeptane for the ase of single-hop and multi-hop alls. Single-hop ase: For the single-hop ase, a all from node j to its neighbor node l is aepted if there is at least one free slot in the region R(j). From our assumption about the fat that the routing protool satisfies the property that at most nodes on the path an hear any other node on the path, we have for a given number of alls in the region R(j) P A ¼ PðNumber of free slots Þ ¼ PðNumber of used slots B Þ Using Lemma P A P Number of alls B Yk¼K q k ðjþ n k GðjÞ n n þn þþn K b B k¼ k! where GðjÞ ¼ P Q k¼k q k ðjþ n k n þþn K B k¼ n k! : For a single-lass of alls P A GðjÞ ðþ b q ðjþ i ; where GðjÞ ¼ i¼b q ðjþ i : i! i! i¼ i¼ ðþ i¼ B Multi-hop ase: Consider the attempt to setup an (M - )-hop all (M C ) along the nodes (p,, p M ). The probability of all aeptane is given by Eq. 8. From Eq. 8 andlemma : P A P Number of alls in Rðp Þ B P Number of alls in R ðp Þ B P Number of alls in R ðp Þ B...P Number of alls in R ðp M Þ B P A Gðp Þ G ðp Þ n þn þþn K n þn þþn K... G ðp M Þ G ðp M Þ b B b B Yk¼K k¼ n þn þþn K n þn þþn K b Yk¼K k¼ b B q k ðp Þ n k n k! q k ðp Þ n k B Yk¼K k¼ n k! Yk¼K k¼ q k ðp M Þ n k n k! q k ðp M Þ n k n k! ðþ

9 Wireless Netw () 6:9 99 where G ðjþ ¼ P Q k¼k n þþn K B k¼ P n þþn K B Q k¼k q k ðjþ n k k¼ n k! : For the single-lass ase: P A Gðp Þ b q ðp Þ i i¼ B i¼ G ðp Þ... G ðp M Þ i! b q ðp Þ i i¼ B i¼ i! b q ðp M Þ i i¼ B G ðp M Þ i¼ i! b q ðp M Þ i i¼ B i¼ i! q k ðjþ n k n k! and G ðjþ ¼ ð6þ Using the same approximations as in Eqs. and, we an determine the minimum value of the aeptane probability P Min A : P Min A ¼ G G P Min A ¼ b i¼ B i¼ b i¼ B i¼ i¼ b B G i¼ q Mini i! q Mini i! q Mini i! for single-hop alls i¼ b B G M i¼ q Mini i! for multi-hop alls ð7þ ð8þ where G ¼ P i¼b q Maxi i¼ i! ; G ¼ P i¼b q Maxi i¼ i!, and G ¼ P i¼b q Maxi i¼ i! : () The Case of Preemption: The analysis so far has been done under the assumption that high-priority alls annot preempt lower-priority ones. However, a realisti senario may require that high-priority alls are ensured high probability of all aeptane. This may require introdution of preemption into the system. The analysis of the steady-state probabilities of a preemptive Markov proess is a diffiult problem. The stationary distribution of the highest priority alls an be easily obtained sine these alls effetively ignore the presene of other low-priority alls. Thus, the stationary distribution of the lass- alls is the same as that of the single-lass system given in Eqs.,, 7, and 8. () System Saturation Probability: For the ase of a single-lass of alls, the probability that the network is saturated i.e., no further alls an be aepted is given by P Sat. If the number of type-v alls in a region is B, then this would require at least B slots to be used, and no further alls an be aepted. PðSaturation in RðjÞÞ ¼ PðB slots are usedþ PðSaturation in RðjÞÞPðNumber of type-v alls at RðjÞ¼BÞ q ;V ðjþ B HðjÞ B! ð9þ P Sat Yi¼N q ;V ðiþ B ðþ HðiÞ B! i¼ " # P Sat Max q B N ;V ðþ H B! () Calls with Varying Bandwidth: We now onsider the ase where the alls have varying bandwidth requirements i.e., in other words, eah all uses up different numbers of slots. The lass of the all is determined by its bandwidth requirement. So we now have K lasses of alls. Class alls require slot, lass alls require slots, lass k alls require k slots, and so on. We assume that the alls of all lasses have equal priority, i.e., no all preemption in the network. The varying number of slots hanges the results presented in Lemmas and. If we were to onsider the type-v alls in a region R(j), a type-v all of lass i,b i B K would onsume at least i slots. Hene, Lemma is replaed by Lemma P(Number of used slots in a region RðjÞxÞP P K i¼ i n i;v x ; where x N: Similarly, Lemma is replaed by Lemma P P K i¼ i n i x PðNumber of used slots inaregionrðjþxþ; wherex N: Plugging these inequalities into the derivation for bounds for the single and the multiple hop ases, we get the following bounds for the all aeptane probability for a lass b all whih requires b slots: P Max A ðbþ ¼ HðbÞ P K B bþ in i¼ i;v B Yk¼K Max q n k;v k;v k¼ n k;v! for single-hop alls ðþ

10 Wireless Netw () 6:9 P Max A 6 ðbþ¼ 6 6 for multi-hop alls where HðbÞ¼ Yk¼K Max q n k;v k;v k¼ n k;v! P Min A ðbþ ¼ P Min GðbÞ A ðbþ ¼ 6 GðbÞ where P P B bþ P K i¼ in i;v B P K B bþ in i¼ i;v B b HðbÞ P K B bþ in i¼ i;v B b P Q k¼k Max q n k;v P k;v n k;v! K in i¼ i;v B k¼ ; and H ðbþ¼ P K in i¼ i b B b 6 G ðbþ 6 G ðbþ P K in i¼ i B b Yk¼K Max q n k;v k;v k¼ H ðbþ n k;v! 7 Yk¼K Max q n k;v k;v k¼ H ðbþ n k;v! 7 Yk¼K Max q n k;v k;v k¼ ;H ðbþ¼ P K in i¼ i;v B b k¼ Yk¼K Min q n k k k¼ P K in i¼ i b B b for multi-hop alls P K in i¼ i B b n k! Yk¼K Min q n k k k¼ n k;v! P 7 M ðþ P K i¼ in i;v B b Yk¼K Max q n k;v k;v n k;v! for single-hop alls n k! P K in i¼ i b B b k¼ P K in i¼ i b B b k¼ GðbÞ ¼ P P K Q k¼k Max q n k k k¼ i¼ in i B 7 Yk¼K Min q n k k n k! Yk¼K Min q n k k n k! 7 7 M ðþ ðþ Q k¼k Max q n k k k¼ n k! ; G ðbþ ¼ n k! ; and G ðbþ ¼ Q k¼k Max q n k k k¼ n k! : Note that these bounds tell us that it is not possible for a all requiring more than B to be setup on a path of length greater than, whih is intuitively obvious. () Some examples: Example Consider a senario where all nodes are within eah others transmission range. Also, there is just a single slot in the TDMA system i.e., B = so that only one lass all an be ative in the system. Also all alls belong to the lass and hene have the same bandwidth requirements. The entire network is now a Markov proess with two states: state in whih there is a all that has been aepted into the system and state in whih there is no all ative in the network. The system moves from state to when a all arrives at any of the nodes this proess is Poisson distributed with mean Nk. We an ompute the probability that system is in eah state. Consequently, the probability that a new all is aepted is P A ¼ Pðstate Þ ¼ þq ; q ¼ Nk l : For a single- ¼ PMin A ¼ þq : hop all, from Eqs. and 7, we get P Max A Example Consider another senario where there are three nodes A, B, and C so that A and C are neighbors of B but A and C are not neighbors. There are slots in the TDMA system. The system behavior an be modeled as a Markov proess with states representing all possible onfigurations of lass alls in the system. In this ase, there are states orresponding to alls, all that oupies a single slot (a all from A or C to B), a single all that oupies slots (a all from A to C), and alls. The resulting transition matrix and steady state probabilities are shown in Table. The probability of aeptane of a single-hop all P A ¼ PðÞþPðÞ ¼ þq : The orresponding bounds from Eqs. and 7 redue to P Min A ¼ and P Max þqþ q A ¼ qðþqþ : It is easy to verify that P Min þqþ q A P A P Max A : Define DðqÞ ¼maxð P A P Min A P A ; PMax A P A P A Þ: We an show that D(q) = O() for q?? and D(q) = o() for q?. The probability of aeptane of a two-hop all P A ¼ PðÞ ¼ðþqÞðþqÞ : The bounds are now PMax A ¼ þqþ q and P Min A ¼ : Again D(q) = O() for q?? and ðþqþ q Þ D(q) = o() for q?. (6) A Summary of the Results: The Eqs.,, 7, and 8 suggest that the all aeptane dereases with system load, this derease being rapid at high loads. For an inoming all s hanes of aeptane to be maximized, Eqs. 6 and 9 suggest that the minimum q(j) aross the network be maximized: this suggests that load-balaning would help improve the aeptane rate. Table Transition matrix and steady state probabilities for the system desribed in Example of Set. B. State Probability k k l k l l ðþqþðþqþ q ðþqþðþqþ q ðþqþðþqþ q ðþqþðþqþ

11 Wireless Netw () 6:9 If all the nodes are within the transmission range of one another (all ommuniations are single-hop), then the upper and lower bounds (Eqs. and 7) onverge with q k ðjþ ¼N k k l : k To ensure that the all aeptane is always above a ertain threshold irrespetive of the load, Eqs. 7 and 8 indiate that the network must be well-provisioned i.e., B must be suffiiently high. The Eqs. suggest that the all aeptane dereases with bandwidth requirement and system load, this derease being rapid at high loads. As boundary ases, the following are seen to hold for the all aeptane rates: As the number of slots inreases, it tends to unity. As the all duration inreases, it approahes zero. (7) The ase when the interferene range is not equal to the transmission range: In this setion, we relax the assumption that the interferene range of a node is equal to the transmission range. Nodes that lie within the transmission range of a node j an send pakets to and reeive pakets from node j.on the other hand, a node that lies in the interferene range of node j but outside the transmission range an disrupt ommuniation involving node j in a slot by transmitting in the same slot, although they typially annot ommuniate reliably with node j itself. We denote the interferene range by Q and the transmission range by R.WhenQ [ R, we now have nodes that lie outside the transmission range of a node j that an interfere with the ommuniation involving node j. This redues the spatial reuse of slots. The derease in spatial reuse is indiated by an inrease in the onstant introdued in Set. A. If the paths hosen for routing are suh that, less than onseutive nodes on the path lie within the interferene range of one another, then the all an be routed along the path using at most slots. Here, we estimate as a funtion of Q and R. Thus, for the more general ase of Q [ R, our earlier analysis an be arried through using the appropriate and by replaing the notion of transmission range with that of interferene range. We maintain the earlier assumption that for three nodes A, B, C that are within distane R of one another, alls will always be routed from A to C along the diret link (A, C) instead of links (A, B) and (B, C). Consider a path onsisting of nodes (,,, /). We denote the distane between two nodes (i, j) as d(i, j). We want to plae these / nodes so that nodes on the path (i, i? ),i [ {,,, / - } an ommuniate, the non-onseutive nodes annot ommuniate, and that any pair of nodes an interfere with one another. To onstrut / suh points, onsider a regular polygon of /? points with the nodes {,,, /} mapped to onseutive points of the polygon (exept the point /? whih has no node mapped to it). Let r be the radius of the irumirle of this polygon. Then the distane between onseutive nodes dði; i þ Þ ¼r sin p /þ R: The distane between non-onseutive nodes dði; jþrsin /þ p Rð þ dþ: The maximum distane between any pair of p nodes r sin /þ b/þ ¼ Q: Combining these three onstraints sinð p /þ Þ sinð p /þ b/þ Þ R Q þ d sinð p /þ Þ sinð p /þ b/þ Þ ð9þ We are interested in the maximum value of /?. This gives us the value of the parameter for the given ratio Q R : To see this, onsider a all whose path inludes the path from to / i.e., the all arrives at node from some node j 6 f; ;...;/g; follows the path {,,, /}, and then is transmitted from node / to some node k 6 f; ;...;/g: Say node reeives the all on slot. Eah of the links (i, i? ), i [ {,,, / - } must use a different slot, say i?, beause these nodes interfere with eah other. Finally node / transmits the all to node k on slot /? due to interferene from the remaining nodes in slots {,,, /}. Thus ¼ arg max / sinð p /þ Þ sinð p /þ b/þ! Rð þ dþ Þ Q ðþ The dependene of on the ratio Q R is shown in Fig. 7. We see from the figure that for the ommon ase when the dði; i þ ÞR; i f; ;...; / g ð6þ Rð þ dþdði; jþq; j 6 fi; i ; i þ g; i f; ;...;/g Rð þ dþdð; /ÞQ ð7þ ð8þ Here d is a small positive onstant and we need Q C R(? d). These onstraints ensure that the onseutive Fig. 7 The dependene of the parameter on the ratio of the interferene range to the transmission range Q R

12 Wireless Netw () 6:9 interferene range is not muh bigger than the transmission range, is at most 6. (8) The failure of shortest-path routing: The analysis tells us that the parameters, the all aeptane probability and the system saturation probability depend on the load on the network, the hopount of the path, and the routing protool. We first look at the performane of shortest-path routing relative to the theoretial guarantees. The routing protool is related to the all aeptane and the system saturation probability through the fator f k (i, j) speified in Eq.. Shortest-path routing: Shortest-path routing omputes the shortest-path between the soure and the destination where the distane refers to the Eulidean distane between the nodes. In a highly dense network, the authors of [] proved that the average path length obtained when shortestpath routing is employed in.9r where R is the radius of the network. This leads to heavier load at the enter region of the network. We simulate shortest-path routing and measure the all aeptane rate. The Figs. 8 and 9 indiate the loading of the enter of the network, and dereasing load away from the enter where the ring an be regarded as a unit of distane from the enter (refer Set. for more details). The Figs. show that the shortest-path routing has a all aeptane rate muh below the theoretial limit. Note that even in Fig., the system has several alls with varying hops, whih would be the ase in a realisti senario. The results shown in Figs. are got by measuring the aeptanes for single-hop, -hop, and -hop alls, respetively. The reason that shortest-path routing performs badly is due to the fat that a majority of the alls are routed through the enter of the network resulting in a high load in the enter. This problem suggests Normalized average f Ring Ring Ring Ring number Ring Fig. 8 The normalized average fration of alls being routed to a node with inreasing distane from the enter for shortest-path routing. The arrival rate is. all per seond at a node Normalized average f Ring Ring Ring Ring number Ring Fig. 9 The normalized average fration of alls being routed to a node with inreasing distane from the enter for shortest-path routing. The arrival rate is all per seond at a node PA PA vs Maximum SP Minimum Fig. Call aeptane of single-hop alls using shortest-path routing versus varying load the use of load-balaning to alleviate the formation of hotspots and to inrease the all aeptane. (C) Deterministi guarantees: Our aim is to ensure that a ertain number of alls in the network an be assured of aeptane. We an do so by pegging these alls at a high priority. Consider the following rank-based priority sheme: (Fig. ) Calls are prioritized aording to the lasses to whih they belong. In addition, alls that belong to the highest priority are further alloated to sub-lasses whih are based on the address or ID of the soure of the all. Further, all admission ensures that only one all of a given sub-lass exists in the system. This implies that a partiular node an originate only one suh highest priority all. Preemption is permitted amongst the sub-lasses themselves so that a high-priority sub-lass has a better hane of aeptane. Hene a senario an be envisaged as follows: the network is deployed in a military senario in

13 Wireless Netw () 6:9 PA Class Class DGL Fig. Rank-based priority sheme PA vs PA = Maximum SP Minimum Fig. Call aeptane of -hop alls using shortest-path routing versus varying load PA PA vs Maximum SP Minimum Fig. Call aeptane of -hop alls using shortest-path routing versus varying load PA = PA_Class <= PA = PA_Class < PA_Class whih the nodes are under the ontrol of various ommuniating offiers. The node ID an be assigned based on the rank of the offier using the node. Calls are prioritized at the time of all admission into various lasses. These alls then have probabilities of aeptane depending on the lass to whih they have been assigned and the network state. In addition, the alls of the highest priority lass are assigned to sub-lasses based on their node ID. Thus, to ensure that the all of the highest-ranking offier (say the General) always gets through, the general s node would be assigned a high-priority node ID. Thus, a set of nodes an be designated to ensure ertain all aeptane. To ensure that these guarantees provided are effetive, we need to estimate the number of alls (whih is equivalent to the number of sub-lasses) for whih ertain all aeptane an be ensured, and the all aeptane for the sub-lasses whih lie outside the former lass. () Deterministi Guarantee Limit: TheDeterministi guarantee limit D refers to the number of sub-lasses of the highest priority lass that an be ensured deterministi all aeptane as outlined at the beginning of this setion. These sub-lasses are referred to as the deterministi sublasses. From Number of alls in RðjÞ¼x ) Number of used slots in RðjÞx ðþ If x ¼ B ; then the number of used slots in R(j) B B. If the total number of sub-lasses in the network ¼ B ; then for every node j, the number of used slots in R(j) B B. Thus, this is the number of sub-lasses that an be definitely aepted by every region of the network at a given time. By alloating a unique set of slots to eah of the B sub-lasses, we an ensure that alls of these sublasses are aepted (of ourse, any lower priority alls may need to be preempted in the proess). Thus, the Deterministi guarantee limit D B : This implies that B sub-lasses an be ensured deterministi all aeptane. However, this being a lower bound it may be possible for some more sub-lasses to be ensured of this deterministi aeptane. Independene of the guarantee limit and mobility: At this point, we also would like to point out the effet of the mobility of the nodes on the limit. The deterministi guarantee limit is independent of the mobility. The set of B sub-lasses f;...; g are ensured of deterministi aeptane even in the fae of node mobility. Mobility in the network leads to path breaks and, subsequent, route reonfiguration attempts. In any suh attempt, the alls belonging to the deterministi sub-lasses retain their priority. Thus, these alls are guaranteed resoures during the reonfiguration. () Probability of aeptane for the probabilisti sublasses: The sub-lasses other than the deterministi sublasses are referred to as the probabilisti sub-lasses. Sine sub-lasses are assigned based on node IDs, there are N sub-lasses, designated {,, N} in dereasing order of

14 Wireless Netw () 6:9 priority. We are onsidering the all aeptane of a all belonging to a sub-lass n [ B (all alls in any of the B sub-lasses f;...; g are of a higher priority than this all and are within the deterministi guarantee limit) at a time t. We denote the probability that a all of sub-lass i exists in the network at time t by p i (t). Let q i (t) = - p i (t). Denote: the aeptane of all of sub-lass n as ACC n, and the number of alls [ sub-lasses {,, n - } as Count(, n - ). As in Eq.,ifCount(, n - ) is less than B ; then for every node j, the number of slots used by alls of these sub-lasses is B B -. All the remaining slots are either free or are used by lower-priority alls whih an be preempted by the all belonging to sublass n. Thus, the all of sub-lass n an be aepted. Thus At time t; Countð; n Þ\ B ) Call of sub-lass n is aepted ðþ P Call of sub-lass n is aeptedjcountð; n Þ\ B ¼ ðþ By denoting the probability of aeptane of the all belonging to sub-lass n at time t as P n (t): P n ðtþ ¼ P ACC n jcountð; n Þ\ B P Countð; n Þ\ B þ P ACC n jcountð; n Þ B P Countð; n Þ B ðþ P n ðtþp Countð; n Þ \ B Y Y ðþ P n ðtþ p l ðtþ q r ðtþ Sf;...;n gjsjbls rf;...;n g S B When the alls at eah node follow an idential probability distribution i.e., p j ðtþ ¼pðtÞ; 8j f;...; Ng; Eq. simplifies to i¼ P n ðtþ b B i¼ n i Load-balaning pðtþ i qðtþ n i We onsider the following strategies for load-balaning: ð6þ Ring-based routing: Ring-based routing [] transfers the load from the enter to the periphery of the network. The sheme makes use of heuristis to balane the load. We define the following terms: The enter node or enter of a network, C, is the node for whih, max 8x ðhcðc; xþþ min 8y ðmax 8z ðhcðy; zþþþ for all nodes x, y, and z in the network. Here HC(a, b) denotes the hopount of the shortest path from node a to node b. Eah node in the network belongs to a Ring denoted by Ring i (r i, r i? ). A Ring is an imaginary division of the network into onentri rings about the enter of the network. The thikness of the ring is given by r i? - r i. A node that belongs to Ring i lies at a distane in (r i, r i? ) from the enter of the network. The load balaning heuristi that we use is a Preferred Outer Ring routing Sheme (PORS) []. In this strategy, traffi generated in a node in Ring i and destined for a node in Ring j must not go beyond the rings enlosed by Ring i and Ring j. Further, the pakets must be preferentially routed through the outer of the two rings. Thus, for nodes belonging to the same ring, pakets must be preferentially transferred in the same ring. For nodes belonging to different rings, all angular transmissions must preferentially take plae in the outer of the two rings while the radial transmissions transfer pakets aross the rings. Thus, PORS affets the hopount while at the same time moving most of the load away from the enter. Bandwidth-limited routing: Bandwidth-limited routing is a more diret form of load-balaning that uses an estimate or measurement of the available bandwidth to selet a path. It differs from the two previous methods (shortest-path routing and PORS) in that it is dynami: onstantly adapting to hanges in the network state. There are two opposing metris that suh a sheme attempts to reonile. It tries to hoose paths with the highest available bandwidth. These paths, usually, tend to be longer than the shortest path. As a result, the available bandwidth of the path, whih is the minimum of the available on the onstituent links, is more likely to derease. The sheme that we use is based on the Shortest-dist (P, n) studies in [6]. We use a variant of this heuristi. The weight for the link (u, v) is weighted by n where B(u, v) is the Bðu;vÞ estimated bandwidth of the link, and n is a weighting fator. We simply estimate this as the minimum of the number of free slots at nodes u and v. The intuition behind this heuristi is that when the links are weighted thus, shortest-path routing will selet a path that minimizes P i¼k i¼ d i n B i where k is the

15 Wireless Netw () 6:9 number of hops, d i is the Eulidean distane of the ith hop, and B i is the estimated bandwidth of the link traversed on the ith hop. This heuristi tends to selet links with high available estimated bandwidth that would also form a short path to the destination. We set the exponent n to for our experiments. Simulation studies To study the atual behavior of the parameters of interest, we built a TDMA based Ad ho wireless network simulator in C??. Our simulator models the wireless system desribed in Set. A and B by taking into aount the broadast nature of wireless medium, in the same way as the well known simulators suh as NS- and QualNet model the wireless networks. Unlike these simulators, our simulator laks any modeling of radio propagation pathloss and fading effets in wireless environment. However, as our objetive in this work is to study the influene of routing protools on the all aeptane rate, whih is a measure of the number of alls that an be admitted into the network, our work does not involve olleting any results after a all is established. The all aeptane measurement is performed by the all admission ontrol module while admitting alls into the network. Hene the results presented in this paper do not get affeted by atual paket exhanges in the network. But metris suh as all drop ratio, paket delivery ratio, and end-to-end delay involve olleting measurements after alls get admitted into the network and hene require sophistiated simulators like NS- and QualNet implement radio propagation models in wireless environment. Call admission involves two steps: finding a path using one of the routing protools disussed and reserving slots along the path. Slot alloation for a partiular all is done in a greedy manner. If at any intermediate node, the number of free slots is found to be inadequate, the all is rejeted. Calls are generated at eah node aording to a Poisson proess and the aepted alls have an exponentially distributed all duration. The nodes are not mobile. The parameters of the simulation are speified in Table. The simulated network has nodes whih are distributed in a uniformly random fashion over a terrain of, m 9, m. Transmission range of a node is m. Interferene range of a node is equal to its transmission range. For eah all the destination node is hosen uniformly at random from all other nodes present in the network. Simulation runs are arried out for seeds and eah simulation run is for a duration of s. For the simulation studies, we vary the load by varying the all arrival rate at eah node. We ompare the all aeptane probabilities for varying values of Table Parameters used in the simulation Parameter q = (Average Call Arrival Rate) 9 (Average Call Duration). In order to ompare the theoretial values and the experimental results, we need to translate the q value to the q Max,V value. Thus, we also measure the average fration of alls that pass through a region. This fator is an indiation of the nature of the routing protool used. We then measure the all aeptane of alls based on their hopount for different routing protools and ompare with the theoretial limits. 6 Simulation results (A) Call aeptane probability: We have ompared the probability of all aeptane of shortest-path routing (SP), Bandwidth-limited routing (BW), PORS, and the theoretial bounds at different values of load (in terms of q). We have also studied the aeptane probability for hopount values of,, and (Figs. 6). In all the results, the all aeptane probability value dereases with an inrease in the network load, as expeted. Further, the urves depiting the all aeptane probability values of SP, BW, and PORS lie within the region surrounded by P Max A and P Min A. PA PA vs Value Number of nodes Number of slots Terrain area, m 9, m Transmission range m Average all duration s Simulation duration s Number of seeds Maximum SP PORS BW Minimum Fig. Variation of Call Aeptane versus q for Single-hop alls

16 6 Wireless Netw () 6:9 PA PA vs Maximum SP PORS BW Minimum Fig. Variation of Call Aeptane versus q for -hop alls Normalized average f SP PORS BW Ring Ring Ring Ring Ring number Fig. 7 The normalized average fration of alls being routed to a node with inreasing distane from the enter. The arrival rate is. all per seond at a node PA PA vs Maximum SP PORS BW Minimum Fig. 6 Variation of Call Aeptane versus q for -hop alls Normalized average f SP PORS BW Ring Ring Ring Ring Ring number Fig. 8 The normalized average fration of alls being routed to a node with inreasing distane from the enter. The arrival rate is all per seond at a node PORS performs only marginally better than shortestpath routing (and in fat worse for single-hop alls) while BW performs signifiantly better. PORS attempts to balane the load impliitly by routing alls to the periphery: this may not be the most effetive strategy beause nodes in one ring an interfere with those in other rings. Also it does not take into aount the fat that a longer path would result in more resoures being onsumed affeting the aeptane rate of alls in the future. This is probably the reason why the single-hop alls have a lower aeptane rate in PORS. BW, by using an expliit bandwidth-based load-balaning is evidently more effetive. To bring out the differene in the performane of the three routing algorithms, we ompute the fration of all generated alls that arrive at a node. We ompute the average of this fration for all nodes that belong to a ring and hene an be onsidered to be at a fixed distane from the enter of the network. The Figs. 7 and 8 plot this average fration (normalized so that the least number is and all the others are divided by this least number) for two different loads on the network. In both ases, shortest-path routing has a high load near the enter. PORS shifts this load to the periphery but inurs the ost of higher path length. BW behaves like shortest-path when the network is lightly loaded but shifts the alls to the periphery with an inreasing load. The differene between the theoretial upper bound and the experimental results is partly the result of the approximations and assumptions used in our model. However, the differene also reflets the inadequay of the existing protools in load-balaning. The inrease in the all aeptane probability of the load-balaning shemes as ompared to shortest-path routing indiates the importane of load-balaning in ensuring better throughput in terms of all aeptane. In fat, load-balaning seems to be an important method of approahing P Max A. The results indiate that an ideal loadbalaning based routing protool an ome lose to the theoretial upper bound. (B) System saturation probability: The variation of the probability of system saturation with load is shown in

17 Wireless Netw () 6:9 7 Fig. 9. This metri remains near zero for moderate-toheavy loads, and takes on an appreiable value only at high values of load. This indiates that system saturation is a rare ourrene for the ommon values of load. Thus, the network rarely enters a state where every new all is rejeted. This also implies that for the ommon values of load, it is always possible to ensure that some fration of the alls are guaranteed aeptane. This fration is based on the values of the probability of all aeptane at that load. (C) Calls with varying bandwidth: We now onsider the ase where the alls have varying bandwidth requirements as disussed in Set. B.. The hannel apaity is Mbps and slottime is ms. We onsider lasses of multimedia alls in the network (i.e., K = ). Class alls require Kbps (i.e., b =, voie alls), lass alls require 8 Kbps (i.e., b =, low-quality video alls), lass alls require 6 Kbps (i.e., b =, medium-quality video alls). The total number of slots available in the Saturation probability Saturation probability vs 6 8 Fig. 9 Variation of Saturation Probability versus q network is, (i.e., B = ). We assume that the load (q) is same for all lasses of alls. We measure the all aeptane probability of alls based on their lass and hopount for SP, PORS, and BW protools at varying values of q and ompare with the theoretial limits. The Figs. 8 show the variation of the all aeptane probability versus load for hopount values of,, and. In all the protools, the all aeptane probability dereases with an inrease in the bandwidth (in terms of number of slots, b) requirement from Class alls to Class alls. As observed from the plots, the all aeptane probability dereases more rapidly for multi-hop alls present in the network. Sine it employs expliit bandwidth-based load-balaning in the network, BW routing protool out-performs other two routing protools. As observed from the plots, the urves depiting the all aeptane probability values for all lasses of alls in the ase of SP, PORS, and BW routing protools lie within the region surrounded by P Max A and P Min A. PA PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=) 6 Fig. Call aeptane probability of Single-hop alls (b = ) versus varying load PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=) PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=).. 6 Fig. Call aeptane probability of Single-hop alls (b = ) versus varying load 6 Fig. Call aeptane probability of Single-hop alls (b = ) versus varying load

18 8 Wireless Netw () 6:9 PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=) PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=).. 6 Fig. Call aeptane probability of -hop alls (b = ) versus varying load 6 Fig. 6 Call aeptane probability of -hop alls (b = ) versus varying load PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=) PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=).. 6 Fig. Call aeptane probability of -hop alls (b = ) versus varying load 6 Fig. 7 Call aeptane probability of -hop alls (b = ) versus varying load PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=) PA.8.6. PA vs Maximum (b=) SP (b=) PORS (b=) BW (b=) Minimum (b=).. 6 Fig. Call aeptane probability of -hop alls (b = ) versus varying load 6 Fig. 8 Call aeptane probability of -hop alls (b = ) versus varying load

19 Wireless Netw () 6:9 9 7 Conlusion and future diretions A realisti analysis of the nature of QoS guarantees is ruial in the design of new protools and the improvement of existing ones to handle the growing diversity of demands on networks. In this paper, we have analyzed a TDMA based Ad ho wireless network. We have derived an upper bound on the probability of all aeptane: a bound that gives us a measure of the number of alls that an be allowed into the network, and a lower bound on the probability of system saturation: a number that indiates the likelihood of the network being unable to aept any further alls. Our analysis takes into onsideration the behavior of the routing protool and the inter-dependene of resoures (time-slots) of neighboring regions in a wireless network. Further, our simulation studies indiate that the set of protools tested fall short of the established bounds. Amongst the three protools ompared, the one that inorporated load-balaning out-performed the shortest-path routing based protool. This learly indiates the importane of load-balaning in the attainment of high network performane, and the provision of better QoS guarantees. We have estimated the deterministi guarantee limit. This limit indiates that it is always possible to ensure QoS guarantees for a ertain sub-lass of alls (the sub-lass being a funtion of the node ID of the soure of the all) irrespetive of the mobility and resoure onstraints of the network. One of the key limitations of our analysis is our assumption that nodes are not mobile. When the nodes are moving, the number of nodes in a given region beomes time-dependent. This in turn is refleted in the fator q beoming time-independent. We would like to study the effet of time-dependene of q on the all aeptane. The experimental studies also need to be extended to ompare other protools to infer the essential and desirable properties of protools that approah optimal-behavior. This will also serve as a guideline in the design of protools that attempt to meet speifi QoS guarantees. Seondly, the urrent bounds that we have derived are not tight. Closing the gap between analysis and simulations will provide further insights into the limits on the apaity of wireless networks. A main sope for future work arises from our onsideration of the all aeptane problem in the TDMA setting. While attempting to provide bandwidth guarantees in wireless networks, two important problems arise. The first is the all aeptane in the presene of ontention for the system resoures. The other important problem is all dropping in the presene of hanges in the onfiguration of the system, due to issues suh as interferene and node mobility. 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20 Wireless Netw () 6:9 Author Biographies S. Sriram obtained his B.Teh. degree in Computer Siene and Engineering in from the Indian Institute of Tehnology (IIT), Madras, India. He is urrently working towards the Ph.D. degree in the department of Computer Siene at the University of California, Berkeley, USA. His researh interests inlude Wireless networks, Distributed systems, Network seurity, and Computational biology. T. Bheemarjuna Reddy reeived the B.Teh. degree in Computer Siene and Engineering from Andhra University, India, in and the M.E. degree in Computer Siene and Engineering from the National Institute of Tehnology (NIT), Rourkela, India, in. He was an inautix dotoral student during 6 in the Department of Computer Siene and Engineering at the Indian Institute of Tehnology (IIT) Madras, India, where he foused on QoS provisioning and Multimedia transport in Ad ho wireless networks. During January 7 Marh 7, he was a Senior Projet Offier at IIT Madras, India. He is urrently a post-dotoral researher at the University of California, San Diego, USA. His researh interests inlude Quality of Servie, Multimedia transport, and Cognitive networking in wireless networks. C. Siva Ram Murthy earned his B.Teh. degree in Eletronis and Communiation Engineering from Regional Engineering College (now National Institute of Tehnology), Warangal, India, in 98, M.Teh. degree in Computer Engineering from the Indian Institute of Tehnology (IIT), Kharagpur, India, in 98, and Ph.D. degree in Computer Siene from the Indian Institute of Siene, Bangalore, India, in 988. He has been with the Department of Computer Siene and Engineering at IIT Madras, sine 988, where he is urrently a Professor. He has held visiting positions at the German National Researh Centre for Information Tehnology (GMD), Germany, University of Stuttgart, Germany, University of Freiburg, Germany, Max Plank Institute for Software Systems, Germany, Swiss Federal Institute of Tehnology (EPFL), Switzerland, University of Washington, Seattle, USA, and University of California San Diego, USA. He is the o-author of the textbooks Parallel Computers: Arhiteture and Programming, (Prentie-Hall of India, New Delhi, India), New Parallel Algorithms for Diret Solution of Linear Equations, (John Wiley & Sons, In., New York, USA), Resoure Management in Real-time Systems and Networks, (MIT Press, Cambridge, USA), WDM Optial Networks: Conepts, Design, and Algorithms, (Prentie Hall, New Jersey, USA), and Ad Ho Wireless Networks: Arhitetures and Protools, (Prentie Hall, New Jersey, USA). His researh interests inlude parallel and distributed omputing, real-time systems, lightwave networks, and wireless networks. He has published over papers in refereed international journals and over papers in refereed international onferenes in these areas. Dr. Murthy is a reipient of Best Ph.D. Thesis Award from the Indian Institute of Siene, Indian National Siene Aademy (INSA) Medal for Young Sientists, and Dr. Vikram Sarabhai Researh Award. He is a o-reipient of Best Paper Awards from the th IEEE International Workshop on Parallel and Distributed Real-Time Systems (WPDRTS), the 6th and th IEEE Annual International Conferene on High Performane Computing (HiPC), and the th IEEE International Conferene on Networks (ICON). He is a Fellow of the Indian National Aademy of Engineering, an Assoiate Editor of IEEE Transations on Computers, and a Subjet Area Editor of Journal of Parallel and Distributed Computing.