E-Diophantine estimating peak allocated capacity in wireless networks

Computer Communications xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Computer Communications

journal homepage: www.elsevier .com/locate /comcom

⇑ Corresponding author.E-mail addresses: perez@neclab.eu (X. Costa-Pérez), wu@neclab.eu (Z. Wu),

mezzavilla@neclab.eu (M. Mezzavilla), r.demarca@comsoc.org (J. Roberto B. deMarca), arauz@ohio.edu (J. Aráuz).

Please cite this article in press as: X. Costa-Pérez et al., E-Diophantine estimating peak allocated capacity in wireless networks, Comput. Commun.http://dx.doi.org/10.1016/j.comcom.2015.01.007

Xavier Costa-Pérez a,⇑, Zhendong Wu a, Marco Mezzavilla a, J. Roberto B. de Marca b, Julio Aráuz c

a NEC Laboratories Europe, Heidelberg, Germanyb Catholic University, Rio de Janeiro, Brazilc School of Information and Telecommunication Systems, Ohio University, USA

a r t i c l e i n f o

Article history:Received 9 March 2014Received in revised form 17 November 2014Accepted 10 January 2015Available online xxxx

Keywords:LTEWireless MANWireless LAN3GPPWiMAX

a b s t r a c t

Wireless networks providing QoS guarantees need to estimate the increase in peak allocated capacitywhen considering admitting a new resource reservation in the system. In this paper we analyze differentavailable approaches to compute this capacity increase and, based on their limitations, propose the E-Dio-phantine solution along with two heuristics of polynomial complexity: E-Diophantine-W and E-Diophan-tine-UW. The properties of the designed algorithms are derived through a mathematical analysis andtheir accuracy and computational load characteristics evaluated in a generic scenario. Complementaryto the generic study, a network performance evaluation comparing the different approaches is conductedusing OPNET’s simulator and considering a realistic wireless network.

Based on our results, the main conclusions that can be drawn are: (i) the larger the degree of flexibilityallowed for defining the resource reservations characteristics, the larger the potential benefit of the E-Diophantine solutions both in accuracy and computational load terms and (ii) for systems supporting alarge number of reservations, the E-Diophantine heuristics can be used to reduce the computational loadfrom exponential to polynomial (cubic) at a low estimation error probability cost.

1. Introduction

Wireless networks are a key element of today’s society to com-municate, access and share information. The ever increasing widerange of services and applications building on wireless networkcapabilities results in a diverse range of Quality of Service (QoS)requirements that needs to be fulfilled to ensure user satisfaction.Third (3G) and Fourth-generation (4G) broadband wireless tech-nologies such as 3GPP Long Term Evolution (LTE) [1], LTE-Advanced [2], IEEE 802.16 [3] and IEEE 802.16m [4] have alreadydefined flexible mechanisms to be able to support a large numberof different QoS requirements. In the Wireless Local Area Network(WLAN) domain, a similar path was followed with the standardiza-tion of IEEE 802.11e [5] and IEEE 802.11-2012 [6]. Complying withthe QoS requirements of granted service demands is mandatory forservice providers and requires accurately estimating the availablesystem capacity when deciding whether new service requestscan be accepted. Precise capacity estimation techniques allow for

the design of efficient admission control algorithms in order tomaximize the utilization of networks while ensuring a satisfactoryQuality of Experience (QoE) for users.

In this paper we propose a peak allocated capacity estimationalgorithm, E-Diophantine, and evaluate its performance. This newalgorithm improves the accuracy of prediction versus complexitytrade off when compared with currently available approaches.The results here presented extend our previous work in [7] and[8] by: (i) analyzing the E-Diophantine complexity as comparedto competing approaches and (ii) designing and evaluating twoheuristics, E-diophantine-W and E-diophantine-UW, that reducethe E-Diophantine computational load from exponential to polyno-mial (cubic) at a low estimation error probability cost.

This article is structured as follows. In Section 2 we review thestandardized QoS resource reservation specifications for wirelessnetworks, propose a common modeling and describe solutionsavailable in the literature to estimate peak allocated capacity.Our proposed E-Diophantine approaches are explained in Sections3 and 4 along with their mathematical foundations and corre-sponding complexity analysis. The allocated capacity estimationaccuracy and computational load of the different solutions is com-pared and a realistic wireless system evaluation performed in Sec-tion 5. Finally, Section 6 contains a summary of our findings andconcludes the paper.

(2015),

Table 2WiMAX QoS parameters per data delivery service.

IEEE 802.16 QoS parameters per data delivery service

Data delivery services UGS ERT-VR RT-VR NRT-VR BE

Min. Resv. Tr. Rate (MRTR) � � � �Max. Sust. Tr. Rate (MSTR) � � � �SDU size �Maximum latency � � �Tolerated jitter � �Traffic priority � � �Req./trans. policy � � � � �

2 X. Costa-Pérez et al. / Computer Communications xxx (2015) xxx–xxx

2. Estimation of peak allocated capacity

Wireless networks support QoS reservation of resources byallowing new flows to apply for admittance in the system throughrequest messages indicating their specific requirements. Suchrequests contain a set of QoS parameters which include differentinformation depending on the service type. In the following wereview different QoS reservation schemes as defined by the pre-dominant wireless technologies in order to find out their common-alities. We take here an industry-driven top-down approachwhere, rather than considering the information we would ideallylike to have for our modeling, we analyze which information isactually available based on the standardized specifications of thewireless technologies under consideration. Note that the focushere is not on performing a comprehensive review of all availablewireless standards but rather on providing some major relevantexamples of diverse wireless technologies with a wide acceptancein the marketplace.

2.1. Reservation information available based on standardizedspecifications

2.1.1. 3GPP LTEWe start our analysis considering 3GPP’s LTE, the wireless cellu-

lar technology becoming predominant worldwide. QoS reserva-tions in LTE’s evolved packet system (EPS) are based on bearerswhich correspond to packet flows established between the packetdata network gateway (PDN-GW) and the mobile stations. Thebearer management and control follows the network-initiatedQoS control paradigm and defines two types of bearers: Guaran-teed bit rate (GBR) and Non-Guaranteed bit rate (Non-GBR). Bear-ers are assigned a scalar value referred to as a QoS class identifier(QCI). Several standardized QCI values with specific characteristicshave been defined to allow for successful multivendor deploymentand roaming. Table 1 summarizes these standardized QCIs asdescribed in [9].

2.1.2. WiMAXSecond, we consider WiMAX networks as the major LTE com-

peting technology. WiMAX supports QoS reservation of resourcesby allowing a new flow to apply for admittance in the systemthrough a Dynamic Service Addition REQuest message (DSA-REQ). Such requests contain a QoS parameter set which includesdifferent mandatory information depending on the data deliveryservice requested in the downlink direction (DL), Base Station(BS) to Subscriber Station (SS), or the scheduling service requestedin the uplink direction (UL). Five different QoS services aresupported: Unsolicited Grant Service (UGS), Extended Real-TimeVariable Rate Service (ERT-VR), Real-Time Variable Rate Service(RT-VR) and Best Effort Service (BE). Table 2 summarizes therequired QoS parameter set per Data Delivery Service accordingto the IEEE 802.16 standard [3]. A similar set of parameters isrequired in the uplink direction.

Table 13GPP LTE standardized QCI parameters.

3GPP LTE standardized QCI parameters

Res. type QCI Min. rate Delay (ms) Loss rate Service example

GBR 1 � 100 10�2 Conv. voice

GBR 2 � 150 10�3 Live streaming

GBR 3 � 50 10�3 Real-time gaming

GBR 4 � 300 10�6 Buffered streaming

Non-GBR 5 100 10�6 IMS signaling

Non-GBR 6 300 10�6 TCP-based apps

Please cite this article in press as: X. Costa-Pérez et al., E-Diophantine estimatinhttp://dx.doi.org/10.1016/j.comcom.2015.01.007

2.1.3. Wireless LANFinally, we consider the Wireless LAN technology as the most

popular wireless technology in homes and hotspots nowadays. InWireless LAN QoS reservation of resources has been also enabledas defined by IEEE 802.11e [5] and 802.11-2012 [6]. These stan-dards introduce the Traffic Specification (TSPEC) mechanism whichdefines a set of mandatory QoS parameters in admissible TSPECsfor different service types. In Table 3 we summarize a subset ofthese admissible TSPECs. Note that we focused in this case in thebaseline IEEE 802.11 standard given that it is the mandatorydefault specification to be implemented by any wireless LANdevice. Relevant extensions of the baseline standard as IEEE802.11aa could be further considered for specific use cases or whenwidely adopted by the market.

2.2. Model for peak allocated capacity estimation

Based on the aforementioned set of parameterized QoS guaran-tees for 3GPP LTE, WiMAX and Wireless LAN, we identify a mini-mum common subset of QoS parameters available based on thestandardized specifications composed by (i) starting time ofgranted QoS reservation request, (ii) required data rate and (iii)periodicity at which the guaranteed data rate should be fulfilled(delay bound). Based on this minimum common subset of informa-tion available in the QoS reservations, in the following we define areservation model which will be used for estimating the peak ofallocated capacity when considering admitting a new reservation.

Let us consider an arbitrary reservation ri for which a minimumset of requirements can be defined for services with QoS guaran-tees as:

� Reservation start time: ti (ms)� Reservation required capacity: Bi (bits)� Periodicity of reserved resources: Ti (ms)

Then, based on these requirements, our QoS reservation modelcan be described as follows: given a reservation starting time ti, acertain amount of capacity Bi (bits) is reserved periodically fortransmitting reservation ri data within a time interval Ti. Basedon this reservation model, a resource reservation request ri canbe expressed as a periodic discrete sequence of Kronecker deltaswith amplitude Bi in the following way

riðtÞ ¼ Bi � dt;tiþn�Ti¼

Bi if t ¼ ti þ n � Ti; n 2 ZP0

0 otherwise

�ð1Þ

Once a new reservation request is received, an allocated capacityestimation algorithm needs to evaluate the impact of accepting it onthe currently existing aggregated allocated capacity peak. Assuminga wireless system with N reservations already granted, we defineAðtÞ as the aggregation (as a function of time) of the N flows alreadyin the system plus the one requesting admittance. See Fig. 1 for agraphical representation of the model for a 5 reservations example.

The objective of our estimation algorithm is thus, starting froma given point of time, t0, find the allocated capacity peak, maxðAðtÞÞ,

g peak allocated capacity in wireless networks, Comput. Commun. (2015),

Table 3IEEE 802.11-2012 subset of parameters in traffic specifications.

IEEE 802.11-2012 subset of parameters in admissible traffic specifications

TSPEC MSDU size Mean rate Delay bound Burst size

Continuous � � �Controlled � � �Bursty �Contention � �

X. Costa-Pérez et al. / Computer Communications xxx (2015) xxx–xxx 3

when considering the new reservation request. This can beexpressed as an optimization problem as follows:

maxt AðtÞ ¼XNþ1

Bi � dt;tiþn�Ti

s:t t P t0

In the next section we will consider different possible approachesavailable in the literature to estimate the peak allocated capacity atany given point of time, maxðAðtÞÞ, according to the reservationmodel defined in Eq. 1 and their dependence with the granularitydefined. We define granularity (Gr) as the minimum unit that canbe used for setting ti and Ti. This parameter is necessary to accountfor the time-slotted scheme used in the real wireless systems underconsideration which define admitted frame/slot durations.

Fig. 1. Illustration of QoS reservation model for a 5 reservations exampleconsidering granularity 1 and identification of reservations needs overlap ðri \ rjÞin time.

2.3. Related work

An extensive body of literature exists in the area of reservedcapacity estimation solutions for admission control purposes inwireless networks. In the following we review some of the mainapproaches related to the objectives of our work which focuseson providing deterministic QoS guarantees based on explicitly sig-naled reservation information. Note that, given the large amount ofrelated works, this summary is not meant to be exhaustive but rep-resentative of the available solutions. Well-known literature notspecifically designed for providing deterministic QoS guaranteesas Effective Bandwidth based approaches, e.g., [10,11] or MeasuredThroughput based solutions, e.g., [12], have been omitted for thesake of brevity.

2.3.1. Throughput-basedThroughput-based approaches, e.g., [13], consider the mean or

peak data rate requirements specified by an application in orderto estimate the peak allocated capacity.

In order to determine maxðAðtÞÞ the Throughput-based approachassumes that all admitted reservations need to be served simulta-neously, i.e., without taking into account the time at which flowsactually need to be served. The following equation correspondsto the Throughput-based approximation of AðtÞ.

AThr ¼XNþ1

Bi ð2Þ

Such an approach requires few computational resources; how-ever, it is neither well suited to take into consideration the band-width-varying nature of typical applications such as video, northe times at which resources are required. Thus, in peak-basedimplementations, the actual available resources are likely to endup underutilized while, in mean-based implementations, QoSguarantees might be jeopardized.

2.3.2. Throughput-based solution complexityThe complexity of this solution increases linearly with the num-

ber of reservations considered and therefore, can be expressed asOðNÞ.

2.3.3. Throughput and Periodicity-basedThroughput and Periodicity-based approaches, e.g., [14,15], take

into account not only the rate requirements of the reservationsbut also the delay constraints and/or periodicity of requestsobtained from a traffic description included in the admittancerequest. These solutions make use of the knowledge of the LeastCommon Multiple (LCM) of the periods of the different accepted res-ervations and consider it for estimating the peak allocated capacity.

This Throughput and Periodicity-based approach obtains an accu-rate solution for maxðAðtÞÞ by computing all values of AðtÞ within aTLCM period. Note that since AðtÞ is composed of N + 1 periodic res-ervations, its period TLCM corresponds to the Least Common Multi-ple (LCM) of the periods of the reservations in the system plus theone under consideration. The following optimization problem for-mulation corresponds to the maxðAðtÞÞ computation in this case.

maxt AðtÞ ¼XNþ1

Bi � dt;tiþn�Tið3Þ

s:t t P t0 ð4Þt 6 TLCM ð5Þ

This approach, although accurate, has a dependence with theLCM of the reservations in the system which, depending on the gran-ularity allowed Gr, might increase exponentially with the numberof reservations and thus, become too expensive in computational

terms. Therefore, such a solution might not be feasible in practiceunless a limitation in the granularity of periods is imposed. Thisissue will be studied in Section 5.1.

2.3.4. Throughput and Periodicity-based solution complexityThe complexity of the Throughput and Periodicity-based algo-

rithm increases according to the number of reservations N þ 1and their corresponding LCM as ðN þ 1Þ � LCM. In the worst case,

the LCM increases with the number of flows as 1Gr

QNþ1i¼1 Ti resulting

in a complexity of O Nþ1Gr

QNi¼1Ti

� �! OððTmaxÞNÞ, where Tmax corre-

sponds to the maximum period of the reservations underconsideration.

2.3.5. DiophantineIn order to remove the LCM dependency of the Throughput and

Periodicity-based approach, another solution is considered basedon Diophantine theory which, in general, deals with indeterminatepolynomial equations and allows variables to be integers only. Inthe rest of the paper this approach will be referred as Diophantineand it has already been considered in [16] as a solution for estimat-ing the peak allocated capacity in wireless networks.

We define this solution as follows. Considering a flow alreadyaccepted in the system described with the resource reservationriðtÞ ¼ Bi � dt;tiþni �Ti

and a new flow requesting admittance character-ized by rjðtÞ ¼ Bj � dt;tjþnj �Tj

, the peak Bi þ Bj, will occur for the set ofni and nj combinations which fulfill

ni;nj 2 ZP0 : ti þ ni � Ti ¼ tj þ nj � Tj� �

Based on the previous result, we define the term diophantine setas that composed of non-negative integer solutions for ni and nj

satisfying the condition in Eq. 6. In order to find this set of solutionsEq. 6 can be expressed as the following linear diophantine equationwith two variables

ni;nj 2 ZP0 : ni � Ti � nj � Tj ¼ tj � ti� �

Then, based on linear diophantine equations theory, we knowthat there will be a set of integer solutions for ni and nj if

tj � ti

d2 Z ð8Þ

where d ¼ gcdðTi; TjÞ and gcd stands for greatest common divisor.When the previous condition holds, the diophantine set of solu-

tions corresponding to a specific pair of reservations can be foundwith the extended Euclidean algorithm which will find a and bsuch that

a � Ti þ b � Tj ¼ d; where a; b 2 Z ð9Þ

The diophantine set corresponding to the intersection of reser-vations i and j, indicated with the subindex ij, can be thenexpressed as follows

nij 2 ZP0 : tij þ nij � Tij� �

; where Tij ¼ lcmðTi; TjÞ ð10Þ

where the smallest ni and nj 2 ZP0 satisfy

ti þ nmini � Ti ¼ tj þ nmin

j � Tj � tij ð11Þ

By applying the Diophantine solution to all pairs of reservationsin the system, as well as to their found diophantine sets in a recur-sive manner, an exact solution for maxðAðtÞÞ can be found which isindependent of the LCM length.

2.3.6. Diophantine solution complexityThis solution requires to compute the gcd for all pairs of

reservations in the system as well as the sets of intersections,

diophantine sets, found. Therefore, the total number of gcd execu-tions in the worst case corresponds to

N þ 1i

� �ð12Þ

Eq. (12) presents a maximum at i ¼ Nþ12

because of its symmetric dis-

tribution and, for a fixed value of N þ 1, it can be expressed as 2Nþ1.Thus, since the number of operations to compute the gcd is upperbounded by log2ðTmaxÞ, where Tmax corresponds to the maximum per-iod of the pairs of reservations or intersections under consideration, thecomplexity of the Diophantine solution is Oð2Nþ1 � log2ðTmaxÞÞ ! Oð2NÞ.

Note that since the computational complexity increases expo-nentially as the number of reservations grows, the Diophantinesolution might become also unfeasible in practice.

3. E-Diophantine

Based on the performance issues identified for the approachesdescribed in the previous section in [7] we proposed an enhance-ment to the Diophantine solution, hereinafter referred as E-Dio-phantine. The objective is to achieve the same accuracy whenestimating the aggregated allocated peak capacity but at a lowercomputational cost. The E-Diophantine concept has been patentedby NEC in multiple countries [17].

The E-Diophantine solution proposed consists in first, exactly as inthe original case, finding the sets of intersections, diophantine sets,between all pair of reservations under consideration. After this step,instead of repeating the process in a recursive manner for all dio-phantine sets found, the results are structured in what we will referto in the rest of the paper as matrix of intersections of reservations.

We define the matrix of intersections as a matrix where the infor-mation regarding the intersections found between all pairs of reser-vations in the system are indicated with a 1 when there is anintersection and 0 otherwise. See Table 4 for a 10 reservations exam-ple where the values for ti and Ti of each reservation were randomlydrawn from a uniform distribution between 1 and 10. Based on thismatrix, the rest of the sets are derived based on the informationobtained regarding the reservations involved in each intersection.

In [7] we provided the theorems and proofs that enable thedescribed E-Diophantine solution. For the sake of brevity this infor-mations is not repeated here. The interested reader is referred to[7] for further details on the analytical demonstration.

3.1. E-Diophantine algorithm

Algorithm 1 details the steps followed by the E-Diophantine solu-tion. The first part of the algorithm, which finds the diophantine sets,is identical to the Diophantine approach. Once the sets have beenobtained, a matrix of intersections is constructed by simply structur-ing the diophantine sets results regarding the intersections betweeneach pair of reservations in a matrix form. This operation corre-sponds to the function compute matrix intersð:Þ in the algorithmand an example is shown in Table 4 for a set of 10 reservations.

The problem of finding the aggregated peak allocated capacityis similar to the clique problem which consists in finding particularcomplete subgraphs (cliques) in a graph, i.e., sets of elements whereeach pair of elements is connected [18,19]. The clique problem isbased on an adjacency matrix where the edges between verticesare given for a specific graph indicating their interconnection.This adjacency matrix can be mapped in our case to the matrixof intersections where edges represent intersections betweenreservations. Our goal is then finding a complete subgraph contain-ing the weighted maximum clique. This problem has been exten-sively researched in the past and several solutions are availablein the literature, see for instance [20].

Table 4Example of matrix of intersections of reservations.

r1 r2 r3 r4 r5 r6 r7 r8 r9 r10

r1 1 0 1 1 0 0 1 0 1 0r2 0 1 1 0 1 0 0 0 1 1r3 1 1 1 1 1 1 1 1 1 1r4 1 0 1 1 0 0 1 0 1 0r5 0 1 1 0 1 1 0 1 1 1r6 0 0 1 0 1 1 0 0 0 0r7 1 0 1 1 0 0 1 0 1 0r8 0 0 1 0 1 0 0 1 1 0r9 1 1 1 1 1 0 1 1 1 0r10 0 1 1 0 1 0 0 0 0 1

The E-Diophantine algorithm designed is based on the gen-eral backtracking solution [21]. It consists in a recursive algo-rithm which incrementally builds, on a reservation byreservation basis, a set of candidate intersections keeping foreach reservation the aggregated peak resource reservation in avector called solutions. The solutions obtained starting the back-tracking process at each different reservation are kept in solu-tions and once all have been explored the functionfind maximumðsolutionsÞ simply selects the maximum value con-tained in this matrix.

In order to improve the efficiency of the backtracking searchð:Þfunction, we introduce the condition weight þ solutionðnextÞ 6 maxin Algorithm 1 where weight corresponds to the aggregate band-width in the current candidate set while max corresponds to theaggregated allocated capacity peak found so far. Based on this condi-tion we can determine if the consideration of a new reservation inour current candidate set will result in a larger aggregated resourcereservation peak. If the condition is true we can skip backtrackingbased on this reservation and move on to the next one. The samelogic applies to the condition weight þmax potentialðstartÞ 6 maxwhere we evaluate at each iteration whether it is still possible forthe current set considered to be above the peak found in previousiterations.

Algorithm 1. E-Diophantine algorithm to find out the peakresource requirement for a new reservation rNþ1 with startingtime tNþ1, period TNþ1 and requirement BNþ1 considering the set ofN reservations already accepted in the system with their corre-sponding starting times t ¼ ðt1 . . . tNÞ, periods T ¼ ðT1 . . . TNÞ andrequirements B ¼ ðB1 . . . BNÞ

1: Call executed for each new reservation request

2: for i ¼ 1 to N þ 1 do3: for j ¼ iþ 1 to N þ 1 do4: if solution existsðti; tj; Ti; TjÞ then5: intersections find inters diophðti; tj; Ti; TjÞ6: end if7: end for8: end for9: m inters compute matrix intersðintersectionsÞ10: m inters orderðm intersÞ11: for root ¼ N þ 1 to 1 do12: solutionsðrootÞ backtracking searchðroot;Broot ;maxÞ13: end for14: return find maximumðsolutionsÞ15: Function backtracking_search(start,weight,max)16: next find next intersðstart;m intersÞ17: if next ¼¼ NULL then18: if weight > max then19: max ¼ weight20: end if21: end if22: while next – NULL do23: if weight þ solutionsðnextÞ � max then24: return25: end if26: if weight þmax potentialðstartÞ � max then27: return28: end if29: backtracking searchðnext;weight þ Bnext;maxÞ30: next find next intersðnext;m intersÞ31: end while32: end function

Then, before starting the backtracking algorithm based on thematrix of reservations, we order the reservations based on theirpotential aggregated peak resource reservation BpotðresvÞ. The oper-ation corresponds to the function orderð:Þ and its advantage is that itincreases the probability of finding the peak solution during the firstbacktracking operations. Thus, reducing the need of backtrackingiterations to complete the full exploration of possible solutions.

Finally, note that the E-Diophantine solution can adapt to enter-ing and leaving reservations by just removing or adding new col-umns/rows in the matrix of intersections. In the case of removingreservations, the only operation needed is the deletion of the cor-responding reservations in the matrix of intersections and the sub-straction of the corresponding reservation from the resourcereservation peak previously computed, if applicable. In the caseof adding a new reservation, our solution requires finding out thepotential intersections between the new reservation and the onesalready present in the system. The rest of the matrix of intersec-tions remains unchanged and does not require new computations.

Fig. 2 illustrates the tree of solutions found based on the matrixof intersections in Table 4 and applying the E-Diophantine algo-rithm. As an example, in the case of reservation 1; r1, there is nobranch of solutions with r2 since according to the matrix of inter-sections r1 does not intersect with r2. In the case of the branch ofsolutions r1 ! r3 ! r4 the continuation with the branch of r5 isalso discarded since reservations r1 and r3 do not intersect with r5.

3.1.1. E-Diophantine solution complexityThe first part of the E-Diophantine algorithm, which finds the

first set of intersections (diophantine sets) between all pairs of res-ervations, requires in the worst case the same number of operationsas the first part of the Diophantine algorithm: Nþ1

� �� log2ðTmaxÞ.

After this, both algorithms differ. In the E-Diophantine case, theset of intersections found are used to construct a matrix of inter-sections. Based on this matrix, the rest of the intersections arederived by exploring the tree of possible solutions, solutions treein Algorithm 1, without requiring the gcd computation.

As aforementioned, the exploration of the rest of possible inter-sections in order to find the peak resource requirement is similar tosolving the weighted maximum clique problem for an arbitrarygraph. In our particular case, this problem can be translated tothe unweighted maximum clique problem (all weights are equal)by defining a reference resource requirement unit Bref and thenmapping the resource reservations requests in the matrix of inter-sections according to B=Bref

. Thus, resulting in an increase of the

number of reservations of:

N þ 1! ðN þ 1Þ� ¼XNþ1

�ð13Þ

ðN þ 1Þ� can be then expressed as ðN þ 1Þ� ¼ k � ðN þ 1Þ, where k is aconstant controlled by the network operator.

Fig. 2. Illustration of tree of solutions of intersections for the E-Diophantinealgorithm based on the matrix of intersections shown in Table 4.

Efficient exact algorithms for the unweighted maximum cliqueproblem have been extensively researched in the literature. Rob-son [22] represents one of the fastest solutions known today which

incurs in a complexity of O 2N4

� �. Considering the complexity of

both parts of the E-Diophantine algorithm together, the overall

complexity can be expressed as O 2k�N4

� �.

The worst case E-Diophantine complexity thus can be poten-tially worse than the Diophantine one depending on the k valuedetermined by the operator configuration. However, in contrastto the generic maximum clique case, in our particular problem wedo not deal with random edges between vertices and this can beexploited to obtain better performance. These differences will beexplained in more detail in the next section and their actual perfor-mance compared in Section 5.1.

4. E-Diophantine heuristics: E-Diophantine-W andE-Diophantine-UW

In the previous section we have shown that the E-Diophantineexact solution might still become unfeasible in practice due to its

exponential growth with the number of reservations. In the follow-ing we describe two non-exact heuristics designed to overcome thislimitation by performing a partial exploration of the tree of solutionsinstead of an exhaustive one. The proposed approaches achieve apolynomial computational load increase (cubic) based on the num-ber of reservations at a bounded probability estimation error cost.

Note that, as explained in Section 3.1, the E-Diophantine exactsolution can be mapped to the unweighted maximum clique prob-lem which is NP-hard and therefore, no polynomial solutions areexpected to be found. The difference of this problem to our partic-ular case is that we can exploit the fact that, in our cliques, we donot deal with random edges between vertices but with edges deter-mined by common properties based on our matrix of intersectionsconstruction.

The E-Diophantine heuristics are based on the following obser-vation: Let us assume that three reservations ri; rj and rk result intwo diophantine sets tij þ nij � Tij

� �and tik þ nik � Tikf g having reser-

vation ri in common. In such a case, since both diophantine sets area contained inside the set ti þ ni � Tif g it can be intuitively observedthat the probability of reservations rj and rk of intersecting witheach other is higher than if they would not have a reservation incommon. Moreover, the larger the number of reservations in com-mon between diophantine sets, the larger the probability of inter-secting with each other. The actual probability though cannot becomputed unless specific assumptions are made on the way thestarting times and periods of reservations are chosen.

4.1. E-Diophantine Weighted (E-diophantine-W)

As aforementioned, a reduction of the computational load of E-diophantine can be achieved by performing a partial exploration ofthe tree of solutions instead of an exhaustive one. The approachtaken in order to select which branches to explore and which onesto discard is the following and is described in pseudo-code inAlgorithm 2.

First, the matrix of intersections is built in the same way asdescribed for the E-Diophantine algorithm. Then, for each reserva-tion we start the process of selecting which branch from the possi-ble solutions tree is evaluated next until no more reservations canbe considered. The selection process consists in first creating, foreach reservation, a candidate list with the rest of the reservationswith which it could potentially intersect and a black list with thereservations which should be discarded. This operation corre-sponds to the function initiateð:Þ. After this, for each reservationin the candidate list the function find next candidateð:Þ computesits Weight considering the potential aggregated peak allocatedcapacity that would be required by this reservation and removingthe reservations present in the black list. Once the Weight for allreservations has been computed, the reservation with the largestWeight is selected as the next branch to be explored and the blacklist updated accordingly by the function updateð:Þ. The process isthen iterated until no more reservations can be considered.

4.2. E-Diophantine Unweighted (E-diophantine-UW)

A complementary solution to E-Diophantine-W can be designedby converting our Weighted Maximum Clique problem to anUnweighted one by applying a transformation to the matrix ofintersections as indicated in Eq. 13. Compared to the Weightedproblem the advantage of this transformation is that our algorithmcan concentrate in finding the branch of the tree of solutions withthe largest length without having to consider the actual Weight ofthem. In the downside though a larger number of reservationsmight need to be considered depending on the k value determinedby the network operator Bref configuration.

We apply this principle to design an alternative to E-diophantine-W which we will refer to in the rest of the paper as E-DiophantineUnweighted (E-diophantine-UW). In this case the same Algorithm 2can be used by transforming the matrix of intersections used asinput according to Eq. 13 and simplifying the find next candidateð:Þfunction such that it neglects the Weight information.

Algorithm 2. E-Diophantine-W algorithm to find out the peakresource requirement for a new reservation rNþ1 with starting timetNþ1, period TNþ1 and requirement BNþ1 considering the set of Nreservations already accepted in the system with their correspond-ing starting times t ¼ ðt1 . . . tNÞ, periods T ¼ ðT1 . . . TNÞ and require-ments B ¼ ðB1 . . . BNÞ

1: Call executed for each new reservation request

2: for i ¼ 1 to N þ 1 do3: for j ¼ iþ 1 to N þ 1 do4: if solution existsðti; tj; Ti; TjÞ then5: intersections find inters diophðti; tj; Ti; TjÞ6: end if7: end for8: end for9: m inters ¼ compute matrix intersðintersectionsÞ10: for i ¼ 1 to N þ 1 do11: initiateðblack list; candidate listsÞ12: while cnt þ sizeðblack listÞ < N þ 1 do13: candidate lists find next candidateðm inters; iÞ14: updateðblack list; candidate listsÞ15: cnt ¼ cnt þ 116: end while17: end for18: return find maximumðcandidate lists;BÞ

1 Note that in general the instant at which a set of users in a system start usingservices (ti; tj , . . .) and the periodicity in the need of resources of each service (Ti;

Tj , . . .) is uncorrelated and thus, the intersection between their reservations as well.

4.3. E-Diophantine heuristics solutions complexity

The first part of the E-Diophantine-W and E-Diophantine-UWalgorithms is identical to the Diophantine one and requires

� �� log2ðTmaxÞ operations to obtain the matrix of intersections.

In addition to this operation, Algorithm 2 explores for each res-ervation (N þ 1) the rest of potential intersecting reservations (N).Then, for each of them a comparison is performed considering thecandidate and black lists (N) in the function find next candidateð:Þ.Thus, the worst case complexity of the optimization algorithms can

be approximated as OðN3Þ for E-Diophantine-W and Oððk � NÞ3Þ forE-Diophantine-UW.

4.4. E-Diophantine heuristics: probability of estimation error

Since the E-Diophantine heuristics described in the previous sec-tion do not perform an exhaustive search through the matrix ofintersections but a partial one, it could happen that these solutionsunderestimate the actual aggregated allocated capacity peak. Inthis section we analyze the probability of this case to happen.We focus in E-Diophantine-UW for analysis simplicity reasons. Asimilar analysis can be applied to the E-Diophantine-W case.

In order to calculate the probability of E-Diophantine-UW select-ing a branch of solutions (selected) different than the one contain-ing the actual peak (max), we consider a matrix of intersectionsof N � N reservations and two possible branches containing differ-ent sets of intersecting reservations Sselected ¼ Is1 ; . . . IsN andSmax ¼ Im1 ; . . . ImN ; where I takes the corresponding binary valueindicating whether an intersection between two flows occur based

on the matrix of intersections. Then, we define the probability of areservation of intersecting with another as Px and i and j as thetotal number of intersections in Sselected and Smax, respectively. Thus,assuming that the intersections between reservations are indepen-dent from each other,1 the probability of the Smax branch containingthe maximum number of intersections, k, instead of Sselected can beexpressed as follows

P�sel¼ ðPxÞ j � ðPxÞ2ðk�1Þ � ðPxÞi

Xi�2

l¼i�kþ1

ð1� PxÞ2l ð14Þ

where i P j P k; Px < 1; i < N and k < NSince our objective is to find an upper bound for the error prob-

ability, Eq. 14 can be simplified by assuming j ¼ k because thelower the value of j, the higher the value of P�sel

P�sel¼ ðPxÞ3kþi�2

Xi�2

l¼i�kþ1

where i P k; j ¼ k; Px < 1; i < N and k < NEq. 15 provides the error probability of selecting a branch from

the tree of solutions that does not contain the maximum. Since anerror when selecting the next branch could occur in k to i branchselection operations in the worst case, the overall error probabilitycan be expressed as

P� ¼Xi

ðPxÞ3kþs�2Xs�2

l¼s�kþ1

where i P k; j ¼ k; Px < 1; i < N and k < NBased on Eq. 16 we plot in Fig. 3 for illustration purposes the

value of P� for a wide range of maximum number of intersections(k) and probabilities of intersections (Px) considering j ¼ k ¼ i. Notethat this case represents an upper bound since the larger the valuesof i and j above k, the lower the value of P�. Moreover, in order tocompute the actual probability of different reservations of intersect-ing with each other, specific assumptions regarding the way theirstarting times and periods would be required. Thus, a sweep on dif-ferent possible values has been performed to cover most cases.

As it can be observed from the figure, the E-Diophantine heuris-tics introduce a low error probability (P� < 0:1%) and present theinteresting property that, as the maximum number of intersectionsincreases (relevant case for admission control decisions), P�decreases, independently of the probability of intersection value.The reason for this property is that as the number of flows in thesystem increases, the number of intersections in the matrix ofintersections also increases and thus, the branch selection processis more accurate. Note that, as explained in Section 4.1, for eachreservation a candidate list with the rest of the reservations withwhich it could potentially intersect is created and the branch withthe largest weight selected. Therefore, the larger the number ofintersections available in the selection process, the lower the prob-ability of selecting a branch not containing the maximum.

5. Performance evaluation and discussion

5.1. Estimation algorithms of peak allocated capacity

In the previous sections we have described different options tocompute the peak allocated capacity based on QoS reservationinformation as defined by the currently most deployed wirelessstandards. In this section we compare the performance of these

2 The Throughput-based case is not considered since its computational load is lowbut its estimation accuracy very poor (see Fig. 4(a)).

3 For instance, in the 30 reservations case with granularity 10, the computationtime in a 2 � Quad Core simulation server took >1000 s.

80100 0.05

0.250.5

0.750.95

5x 10−3

Probability of intersection (Px)Maximum number of intersections (k)

x 10−3

Fig. 3. Illustration of the error probability according to Eq. 16 for j ¼ i ¼ k and awide range of Px and k values.

algorithms both in estimation accuracy and computational loadterms to provide an insight on the trade-offs to be consideredwhen deciding which option is more appropriate for a specific net-work deployment.

In order to evaluate the performance differences between theThroughput-based, Throughput and Periodicity-based, Diophantineand E-Diophantine approaches, we implemented these algorithmsin Matlab and designed the following experiment. We considereda system with 10–100 reservations where, for each one, ti and Ti

were randomly drawn from a uniform distribution ranging fromthe granularity value chosen up to 100 (in multiples of the granu-larity). Three different granularity values were evaluated: 1, 5and 10. Bi was randomly drawn from a uniform distribution rang-ing from 1 to 100. Fig. 4 summarizes the results of the experimentafter running a minimum of 100 seeds for each value. Note that theconfidence intervals are not displayed in this case because theirsuperimposition with the symbols of the different approacheswould significantly degrade the readability of the figures.

In Fig. 4(a) the difference between the estimated maximumnumber of resources required by each of the approaches can beobserved. Taking the Diophantine results as reference since it rep-resents the exact solution, as expected, the Throughput-basedapproach is the one presenting the largest difference to the actualvalues; reaching differences of more than 100% in some of thecases. Such a large estimation deviation to the actual value wouldobviously result in available resources being underutilized andthus, in a lower revenue for a network operator.

In the Throughput and Periodicity-based case, the smaller thegranularity, the larger the difference to the correct value due to alimitation in the maximum TLCM value that can be considered in areal implementation (107 in our system). Furthermore, the estima-tion is below the actual value and therefore, its usage for admissioncontrol purposes could compromise the QoS guarantees. Regardingthe E-Diophantine estimation, as expected, it is always equal to theexact value (Diophantine). Finally, the E-Diophantine heuristicspresent a different accuracy depending on the solution considered.While E-Diophantine-W closely matches the correct prediction, E-Diophantine-UW overestimates because of the Bref chosen (25 in thisexperiment). This Bref value was chosen as an intermediate value toillustrate the trade-off to be considered when configuring the E-Diophantine-UW algorithm. The larger the Bref value, the smallerthe computational load increase as compared to the E-Diophan-tine-W solution but at the cost of accuracy loss.

Regarding the computational complexity, Table 5 summarizesthe worst case complexity of the different solutions considered

as derived in Sections 2 and 3. Since the upper bounds providedin Table 5 correspond to the worst case performance, in this exper-iment we analyze the differences to be expected when consideringa wide range of possibilities regarding the number of flows in thesystem. In Fig. 4(b) the computational load differences are shown,computed as the percentage of reduction achieved with respect tothe Throughput and Periodicity-based approach, taken here as refer-ence due to its implementation simplicity.2

As it can be observed, the Diophantine solution, although exact,exceeds by far the computational load of the alternative solutionsconsidered and thus, it might not be feasible in practice.3 For thelargest granularity considered (granularity 10), the Throughput andPeriodicity-based approach clearly outperforms in computational timethe E-Diophantine solutions with no loss of accuracy. However, as thegranularity considered decreases, the E-Diophantine solutions reduc-tion with respect to the Throughput and Periodicity-based oneincreases. In particular, the E-Diophantine algorithm presents the larg-est computational load advantage as compared to the Throughput andPeriodicity-based but this advantage eventually starts decreasing asthe number of reservations grows. With respect to the E-Diophantineheuristics, while E-Diophantine-W manages to keep a very consider-able advantage even for a large number of reservations (100), theE-Diophantine-UW advantage starts lower and eventually gets outper-formed by the Throughput and Periodicity-based algorithm. The reasonis the increase in the number of reservations to be considered by theE-Diophantine-UW approach which grows according to Eq. 13.

Based on these results, we can conclude that the E-Diophantine-W approach is, among the considered options, the most appropri-ate solution when fine granularities for resource reservations areconsidered. Note that the larger the granularity considered, thecoarser the peak capacity estimations and thus, more radioresources can be left potentially unused due to conservative esti-mations. As an example, LTE considers subframes of 1 ms duration.

5.2. Wireless network performance evaluation

In the previous section we have analyzed the performance ofthe proposed E-Diophantine solution as compared to its alterna-tives considering a generic scenario. In this section, we completethis evaluation by considering additional elements in the perfor-mance comparison that could have an impact in the allocated peakcapacity estimation of the different approaches. Examples of theseelements are: wireless physical channel, Transport layer, Networklayer, MAC layer, control plane signaling, realistic applications, QoSscheduler, number of stations, etc. In order to achieve this, fromthe different wireless technologies mentioned in Section 2 sup-porting services with QoS guarantees, WiMAX is selected as a rep-resentative example and OPNET’s simulator [23] as evaluation toolfor analyzing an IEEE 802.16 system. The E-Diophantine solutionused is E-Diophantine-W because of its better scalability properties.

An 802.16 simulation scenario is setup according to Table 6consisting of one Base Station (BS) and multiple Subscriber Stations(SS) where each station is configured to send and receive trafficfrom their corresponding pair in the wired domain of its type ofapplication, i.e., one station sends and receives Voice traffic (with-out silence suppression), a second station sends and receives Voicetraffic (with silence suppression), a third one receives a Videostream, a fourth one does an FTP download and the last one doesWeb browsing. The number of subscriber stations is increased inmultiples of five stations up to 125 in total, always keeping therelation of 1=5 of stations of each application type. The QoS

10 20 30 40 50 60 70 80 90 100

Number of Reservations

Granularity 10Granularity 5Granularity 1

ThroughputThroughput & PeriodicityDiophantineE−DiophantineE−Diophantine−UWE−Diophantine−W

(a) Peak of resources allocated for services with QoS guarantees

10 20 30 40 50 60 70 80 90 100

−1e+006

−1e+005

−10000

−1000

Number of Reservations

ty−b

Granularity 10Granularity 5Granularity 1

Throughput & PeriodicityDiophantineE−DiophantineE−Diophantine−UWE−Diophantine−W

(b) Relative computational time of the different estimation algorithms

Fig. 4. Comparison of maximum allocated capacity estimation accuracy and computational load of Throughput-based, Throughput- and Periodicity-based, Diophantine andE-Diophantine approaches.

Table 5Worst case complexity of the different solutionsconsidered.

Complexity

Throughput-based OðNÞThroughput and Periodicity-based OððTmaxÞNÞDiophantine Oð2NÞE-Diophantine O 2

k�N4

� �E-Diophantine-W OðN3ÞE-Diophantine-UW Oððk � NÞ3Þ

scheduling policy chosen is Strict Priority applied first to fulfill theMinimum Reserved Traffic Rates (MRTRs) and then, the MaximumSustained Traffic Rates (MSTRs). The length of the simulations per-formed is 120 s with a warm-up phase of 10 s. In the case of thedelay performance metric, the values represent the 95th percentileof the delay (CDF95) considering all simulation runs. Regarding theconfidence intervals, they are shown only in the cases where theirsuperimposition with the symbols of the different approaches doesnot degrade their readability. The configuration used for the differ-ent applications is detailed below:

� Voice G.711 Voice codec. Data rate: 64 kb/s. Frame length:20 ms. Mapped to the UGS service in the downlink (BS ! SS)and uplink direction.� VAD (Voice with Activity Detection) G.711 Voice codec. Data rate:

64 kb/s. Frame length: 20 ms. Talk spurt exponential with mean0.35 s. Silence spurt exponential with mean 0.65 s. Mapped toERT-VR in the downlink and to ertPS in the uplink.� Video MPEG-4 real traces [24]. Target rate: 450 kb/s. Peak:

4.6 Mb/s. Frame generation interval: 33 ms. Mapped to RT-VRin the downlink and to rtPS in the uplink.� FTP Download of a 20 MB file. Mapped to NRT-VR in the down-

link and to nrtPS in the uplink.� Web Browsing Page interarrival time exponentially distributed

with mean 60 s. Page size 10 KB plus 20–80 objects of a size uni-formly distributed between 5 KB and 10 KB [25]. Mapped to theBE service both in the downlink and uplink direction.

5.2.1. IEEE 802.16: system performance resultsThe objective of the results presented in this section is to

corroborate that the E-diophantine solution successfully predicts

the peak allocated capacity according to the configured MRTRs.Once the agreed aggregated MRTR capacity cannot be served any-more for all classes, a sharp increase in the delay of the differenttraffic classes is expected to be observed according to their actualpriority.

In Fig. 5(a) we show the peak and average throughput experi-enced in the downlink by the different application types as com-pared to the peak capacity estimations of the different approachesdescribed in Sections 2 and 3. The throughput of the differents appli-cations is aggregated according to whether it is Premium traffic(UGS + ERT-VR + RT-VR) or Regular traffic (NRT-VR + BE).

From the performance results in Fig. 5(a) the first remarkableresult is that the peak of Premium traffic is in some cases abovethe peak estimated with the algorithms considered but theThroughput-based one. The reason for this result is the 2 Mb/s MSTRconfigured for RT-VR which allows video applications to get morethan its 500 Kb/s MRTR if there is leftover capacity after serving allMRTRs. Note that the Throughput-based estimation is too conserva-tive and therefore, it will not be considered in the remainder of thissection. Also note that the difference between the Throughput andPeriodicity-based and Diophantine result is caused by the LCM upperlimit configured for the former, set to half of the actual requiredone in order to illustrate its effect on the estimation accuracy.

As the number of stations increases, the difference between theallocated peak capacity estimations and the throughput peak ofPremium traffic decreases. Note that the larger the amount of Pre-mium traffic in the network, the lower the opportunities to goabove the MRTR value. Eventually a point is reached where eventhe MRTR guarantees cannot be satisfied, see crossing pointbetween 100 and 125 stations. Moreover, as the number of flowsin the system increases, the signaling overhead required for theDL-MAP increases as well, resulting in a lower Premium averagethroughput. For illustration purposes, an additional E-diophantinecase has been added, E-Diophantine (1 Mb/s), where the MRTR forRT-VR has been configured to 1 Mb/s instead of 500 Kb/s. This caseprovides an example of how the allocated capacity peak estimationwould vary by allowing bursty traffic to transmit significantlyabove their average.

With respect to the delay performance, the results are shown inFig. 5(b). As expected, when the wireless resources become scarce,the delay experience degrades according to the traffic priority. Inthe case of RT-VR traffic, in contrast to UGS and ERT-VR, the delayexperienced increases constantly. This is due to the performancemetric chosen, 95% percentile of the delay (CDF95), which yields

Table 6Performance evaluation parameters.

IEEE 802.16 configurationBase freq. (GHz) 2.5 DL Subfr. # Subch. 30Bandwidth (MHz) 10 UL Subfr. # Subch. 35Frame duration (ms) 5 # Data Subc./Subch 24Symbol duration (ls) 102.86 # SSs 64 QAM (3/4) 60%Number of subcarriers 1024 # SSs 16 QAM (3/4) 30%DL Subfr. # symbols 35 # SSs QPSK (1/2) 10%UL Subfr. # symbols 12 # SSs per scenario 25–125Multipath channel ITU Ped-B Pathloss model Erceg

Data delivery services Scheduling servicesUGS MRTR 80 Kb/s UGS MRTR 80 Kb/s

Max. Lat 20 ms Max. Lat 20 msERT-VR MRTR 80 Kb/s ertPS MRTR 80 Kb/s

Max. Lat 20 ms Max. Lat 20 msRT-VR MSTR 2 Mb/s rtPS MSTR 800 b/s

MRTR 500 Kb/s Max. Lat 33 msMRTR 800 b/s Max. Lat 20 ms

E-Diophantine configurationUGS BUGS 1600 bits UGS BUGS 1600 bits

TUGS 20 ms TUGS 20 msERT-VR BERT 1600 bits ertPS BERT 1600 bits

TERT 20 ms TERT 20 msRT-VR BRT 16,500 bits rtPS BRT 800 bits

TRT 33 ms TRT 1 s

25 50 75 100 1250

Total Number of Stations

Premium (UGS+ERT−VR+RT−VR) − PeakPremium (UGS+ERT−VR+RT−VR) − AverageRegular (NRT−VR+BE) − Average

ThroughputThroughput & PeriodicityDiophantineE−DiophantineE−Diophantine(1Mb/s)

(a) Downlink Throughput

25 50 75 100 1250

Total Number of Stations

UGSERT−VRRT−VRNRT−VRBE

25 50 75 100 1250

(b) Downlink Delay

Fig. 5. IEEE 802.16 scenario: network performance.

a close to worst case delay for each application traffic and thus, asthe number of flows grows, it increasingly represents the Videopeaks that cannot be absorbed because there is not enough remain-ing capacity after serving all MRTRs. The delay performance of BE,which increases very rapidly, is due to the simple QoS schedulingpolicy used, Strict Priority, resulting in BE traffic being served onlyif the rest of the available traffic has already been served. Finally,the NRT-VR delay performances experience an extreme degrada-tion after the 100 stations point. Note that this is where the peakestimation of the different algorithms but the Throughput-basedcrosses the Premium peak throughput and therefore, the probabilityof NRT-VR traffic to be served decreases significantly.

The performance results corresponding to the uplink directionhave been omitted due to space restrictions since in this case thereis always enough capacity to satisfy the needs of all applicationflows.

6. Summary and conclusions

Wireless networks providing QoS guarantees require an algo-rithm to estimate the increase in aggregated peak allocated capacity

if a new resource reservation is admitted in the system. In this paperwe have proposed the E-Diophantine solution, along with its math-ematical foundations, and evaluated its benefits as compared toalternative approaches of increasing complexity, namely: Through-put-based, Throughput and Periodicity-based and Diophantine. Theperformance comparison comprised both accuracy and computa-tional load analysis in a generic scenario as well as an evaluationusing OPNET’s simulator in a realistic wireless network.

The main conclusions that can be drawn from our results are: (i)E-Diophantine solutions can be successfully used to predict the allo-cated peak capacity demand of admitted QoS reservations in wire-less networks, (ii) the Throughput and Periodicity-based approachcan outperform E-Diophantine in computational terms if limitationsin the period between resource allocations can be imposed, (iii) thelarger the degree of flexibility allowed for defining the resource res-ervation periods, the larger the potential benefit of the E-Diophan-tine solutions both in accuracy and computational load terms and(iv) for systems supporting a large number of reservations, theE-Diophantine heuristics can be used to reduce the computationalload from exponential to polynomial (cubic) at a low estimationerror probability cost.

References

[1] Third Generation Partnership Project (3GPP), Evolved Universal TerrestrialRadio Access (E-UTRA) and Evolved Terrestrial Radio Access Network (E-UTRAN), Overall Description Stage 2, 3GPP TS 36.300 v 9.10.0, January 2013.

[2] Third Generation Partnership Project (3GPP), Requirements for FurtherAdvancements for Evolved Universal Terrestrial Radio Access (E-UTRA, lte-advanced, 3GPP TR 36.913 v 11.0.0, September 2012.

[3] IEEE 802.16 Working Group, IEEE Standard for Local and Metropolitan AreaNetworks, Part 16: Air Interface for Broadband Wireless Access Systems, IEEEStandard 802.16-2009, May 2009.

[4] IEEE 802.16 Working Group, IEEE Standard for Local and Metropolitan AreaNetworks, Part 16: Air Interface for Broadband Wireless Access Systems,Amendment 3: Advanced Air Interface, IEEE P802.16m-2011, May 2011.

[5] IEEE 802.11 Working Group, Wireless LAN Medium Access Control (MAC) andPhysical Layer (PHY) Specifications, Amendment 8: Medium Access Control(MAC) Quality of Service Enhancements, IEEE Standard 802.11e, November2005.

[6] IEEE 802.11 Working Group, IEEE Standard for Local and Metropolitan AreaNetworks, Part 11: Wireless LAN Medium Access Control (MAC) and PhysicalLayer (PHY) Specifications, IEEE Std 802.11-2012, March 2012.

[7] X. Pérez-Costa, M. Mezzavilla, J.R.B. de Marca, J. Aráuz, E-diophantine: anadmission control algorithm for WiMAX networks, in: Proceedings of IEEEWireless Communications and Networking Conference (WCNC), Sydney,Australia, April 2010.

[8] V. Sciancalepore, X. Costa-Pérez, A. Capone. RIA-ICCS: Intercell CoordinatedScheduling Exploiting Application Reservation Information. IEEE WirelessCommunications and Networking Conference (WCNC), April 2013.

[9] Third Generation Partnership Project (3GPP), Technical Specification GroupServices and System Aspects, Policy and Charging Control Architecture, 3GPPTS 23.203 v. 9.13.0, September 2013.

[10] A.I. Elwalid, D. Mitra, Effective bandwidth of general Markovian traffic sourcesand admission control of high speed networks, IEEE/ACM Trans. Netw. 1 (3)(2002).

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[12] S. Jamin, S.J. Shenker, P.B. Danzig, Comparison of measurement-basedadmission control algorithms for controlled-load service, in: Proceedings ofthe IEEE International Conference on Computer Communications (INFOCOM),Kobe, Japan, April 1997.

[13] H. Wang, W. Li, D.P. Agrawal, Dynamic admission control and QoS for 802.16wireless MAN, in: Proceeding of Wireless Telecommunications Symposium(WTS), Pomona, USA, April 2005.

[14] C. Ciconetti, L. Lenzini, E. Mingozzi, G. Stea, Design and performance analysis ofthe real-time HCCA scheduler for IEEE 802.11e WLANs, Elsevier Comput. Netw.J. (CN) 51 (9) (2007).

[15] S. Chandra, A. Sahoo, An efficient call admission control for IEEE 802.16networks, in: Proceedings of the 15th IEEE Workshop on Local andMetropolitan Area Networks, Princeton, USA, June 2007.

[16] O. Yang, J. Lu, Call admission control and scheduling schemes with QoSsupport for real-time video applications in IEEE 802.16 networks, IEEE J.Multimedia (May) (2006).

[17] X. Pérez-Costa, M. Mezzavilla, R. de Marca, A Method for Controlling theAdmission of a Flow to a Network, WO2010EP05617 20100914, 2011.

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E-Diophantine estimating peak allocated capacity in wireless networks

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