Performance analysis of GTS allocation in Beacon Enabled IEEE

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1 1 Performance analysis of GTS allocation in Beacon Enabled IEEE Pangn Park, Carlo Fischione, Karl Henrik Johansson Abstract Time-critical applications for wireless sensor networks (WSNs) are an important class of services spported by the standard IEEE Control, actation, and monitoring are all examples of applications where information mst be delivered within some deadline. Understanding the delay in the packet delivery is fndamental to assess performance limitation for the standard. In this paper we analyze the garanteed time slot (GTS) allocation mechanism sed in IEEE networks for timecritical applications. Specifically, we propose a Markov chain to model the stability, delay, and throghpt of GTS allocation. We analyze the impact of the protocol parameters on these performance indexes. Monte Carlo simlations show that the theoretical analysis is qite accrate. Ths, or analysis can be sed to design efficient GTS allocation for IEEE Keywords: IEEE standard, GTS allocation, Wireless Sensor Network, MAC, Markov Chain. I. INTRODUCTION To spport time-critical applications, IEEE offers a garanteed time slot (GTS) allocation mechanism at the network coordinator. The primary goal of GTS allocation is providing commnication services to time critical data, i.e., make certain garantees on evental delivery and delivery times of packets to be transmitted by local devices to the network coordinator. Specifically, in IEEE , packets are transmitted on a sperframe basis (see Fig. 1). Each sperframe is divided into Contention Access Period (CAP), where nodes contend among each other to send packets, and a Contention Free Period (CFP), where nodes have GTSs to send packets withot contention and ths with garanteed transmission [1]. Most of the papers from the literatre do not consider satisfactorily the CFP, where the GTS mechanism operates. Both CAP and CFP have been stdied in [] and [3], bt the approach is mainly based on simlations. Some interesting algorithm are proposed to improve the performance of GTS allocation mechanism. To imize the bandwidth tilization, the smaller slot size and offline message schedling algorithm are proposed in [4], [5] and [6], respectively. In [7], the delay constraint and bandwidth tilization are considered to design the GTS schedling algorithm. Hang [8] proposes an adaptive GTS allocation scheme by considering the low delay and fairness. An interesting theoretical performance evalation of the GTS allocation has been proposed by Kobaa et al. [9], [1], [11] by sing network calcls. These papers focs on the impact of the IEEE standard parameters (the All the athors are with the ACCESS Linnaes Center, Electrical Engineering, Royal Institte of Technology, Stockholm, Sweden. s: {pgpark,carlofi,kalle}@ee.kth.se. The work was spported by the EU integrated proect SOCRADES, the Swedish Research Concil, the Swedish Strategic Research Fondation, and the Swedish Governmental Agency for Innovation System. beacon CAP >= amincaplength abaseslotdration SO CFP SD = abasesperframedration SO GTS GTS GTS Inactive period = abasesperframedration BO Fig. 1. Sperframe strctre in IEEE beacon beacon and sperframe orders [1]), the delay, throghpt and energy consmption of GTS allocation. In [1], a rond-robin schedler is proposed to improve the bandwidth tilization based on network calcls approach. Network calcls, however, assmes a continos flow model (whereas commnication happens throgh low data rate packets in reality) and it analyzes the worst-case of traffic flows (which leads to severe nder-tilization of time slots in actal environments). Conseqently, the difference between the network flow model of the network calcls approach and the actal behavior may be qite large. In this paper, we focs mainly on the plink scenario, which is the most relevant for sensor networks application, and we develop a theoretical analysis of the impact of GTS allocation mechanism on stability, delay and throghpt. More specifically, the original contribtions of this paper are the following: We present a Markov chain model to analyze the performance of GTS allocation mechanism for IEEE in terms of stability and delay in the slot assignment, and throghpt garanteed. The stability analysis, which gives the qee overflow probability, is qite sefl for nderstanding the stability of the qee size of the network coordinator. Frthermore, we derive these performance measres as explicit fnction of the nmber of reqests arriving dring the sperframes and protocol parameters. We believe that or investigation is the first one to provide sch an accrate modelling and performance analysis. As a reslt, or theoretical analysis can be effectively sed to spport efficiently real-time applications by the optimization of the GTS allocation. The remainder of this paper is organized as follows. In section II, we describe the IEEE standard and the GTS allocation mechanism. In section III, we propose a Markov chain modelling of the GTS allocation. In section IV, we analyze the GTS stability, delay, and throghpt. In section V, we validate the theoretical analysis by simlations, and give performance of GTS allocation. Section VI concldes the paper. II. SYSTEM MODEL In this section we give an overview of the key points of IEEE that are needed for or analysis.

2 The IEEE [1] standard specifies the physical layer and the MAC sb-layer for Low-Rate Wireless Personal Area Networks. The IEEE spports beacon enabled and non-beacon enabled modes. The model selection is decided by the Personal Area Network coordinator (PANC). Fig. 1 shows a sperframe strctre of the beacon enabled mode. The PANC periodically sends the beacon frames in every beacon interval () to identify its PAN and to synchronize devices that commnicate with it. The PANC and devices can commnicate dring active period, called the sperframe dration (SD), and enter the low-power mode dring the inactive period. The strctre of the sperframe is defined by two parameters, the beacon order (BO) and the sperframe order (SO), which determine the length of the sperframe and its active period, respectively. The length of the sperframe () and the length of its active period (SD) are then defined as = abasesperframedration BO, (1) SD = abasesperframedration SO, () where SO BO 14. The abasesperframedration and abaseslotdration denote the minimm length of the sperframe and the nmber of symbols forming a sperframe slot, when BO is eqal to, respectively. The active period is divided into 16 eqally sized time slots. Each active period can be frther divided into a CAP and an optional CFP, composed of garanteed time slots. The slotted or nslotted CSMA/CA is sed within CAP dependent on the beacon enabled and non-beacon enabled mode, respectively. The GTS allocation mechanism of IEEE deals only with the beacon enabled mode. Fig. (a) shows the data transmission dring the CAP of beacon enabled mode. A slotted CSMA/CA mechanism is sed to access the channel of non-time critical data frames and GTS reqests dring the CAP. In the CFP, the dedicated bandwidth is sed for time critical data frames. Fig. (b) illstrates the GTS allocation mechanism within CFP of beacon enabled mode. The PANC is responsible for the GTS allocation and determines the length of the CFP in a sperframe. To reqest the allocation of a new GTS, the device sends the GTS reqest command to the PANC. The PANC confirms its receipt by sending an acknowledgment frame within CAP. Upon receiving a GTS allocation reqest, the PANC checks whether there are sfficient resorces and, if possible, allocates the reqested GTS. The GTS capacity in a sperframe satisfies the following reqirements: 1) The imm nmber of GTSs to be allocated to devices is seven, provided there is sfficient capacity in the sperframe. ) The minimm length of a CAP is amincaplength. Therefore the CFP length depends on the GTS reqests and the crrent available capacity in the sperframe. If there is sfficient bandwidth in the next sperframe, the PANC determines a device list for GTS allocation based on a first-come-first-served (FCFS) fashion. Then, the PANC incldes the GTS descriptor which is the device list that obtains GTSs in the following beacon to annonce the allocation information. The PANC makes this decision within agtsdescpersistencetime sperframes. Note that on receipt PAN coordination CAP Beacon (with CAP length) Data Device Acknowledgement (optional) PAN coordination Device (a) Data transmission in CAP. (b) Data transmission dring CFP. Fig.. Data transfers dring the CAP and CFP of beacon enabled PAN coordinator. CAP CFP GTS reqest Data Acknowledgement Beacon (with GTS descriptor) Acknowledgement (optional) of the acknowledgment to the GTS reqest command, the device contines to track beacons and wait for at most agts- DescPersistenceTime sperframes. A device ses the dedicated bandwidth to transmit the packet within the CFP. In addition, a transmitting device ensres that its transaction is complete one interframe spacing (IFS) period before the end of its GTS. The MAC sblayer needs a finite amont of time to process data received by the PHY. Two sccessive frames transmitted from a device are separated by at least an IFS period. Depending on the length of the transmitted data frames, if the length of frame is less than a amaxsifsframesize then a Short Inter- Frame Spacing (SIFS) period of at least macminsifsperiod symbols will follow. Otherwise Long Inter-Frame Spacing (LIFS) period of at least macminlifsperiod symbols will be sed. In the following section, we propose an analytical modelling of the GTS allocation described above. III. MODELLING OF GTS ALLOCATION Consider the IEEE standard with a star network and a set of N nodes within the PANC s radio coverage. Assme that the network operates in beacon enabled mode. Each device in the range of the PANC generates data packets to be sent to the PANC and informs the coordinator on the need of GTS resorces by sending the reqest dring CAP. Therefore, the PANC needs to allocate a nmber of GTSs by considering the received reqests. These reqests are stored in a qee of the PANC, and wait to be served in the next sperframes, where the related GTS may be allocated. If too many reqests arrive with respect to the PANC qee size, then we have a qee overflow. We consider only the transmit GTSs for the plink traffic. Frthermore, we assme that all GTS transmissions are sccessfl. Each device is allocated at most one GTS and the imm nmber of GTSs in a sperframe is considered according to the IEEE specifications [1]. The nmber of reqests of GTSs depends on the nmber of time critical data packets that the devices want to send to the PANC. With contention-based transmissions dring CAP, there will be loss of GTS reqests de to data collisions in CAP. Hence, the performance of the GTS allocation mechanism in CFP is dependent on the nmber of reqests that arrive sccessflly to the PANC dring the CAP. Sch a nmber is modelled by sing a probability distribtion, which abstracts the contention-based transmissions dring CAP and possible collisions. We will analyze performance obtained by assming different distribtions (Poisson, Normal and Gamma). If there

3 are no sfficient GTS resorces for the reqest in the crrent sperframe, the GTS allocation will be delayed in the next sperframe. For frther details see [1]. The modelling of the GTS allocation is given in two steps. First, we derive the constraints on the nmber of time slots to allocate by considering the details of the GTS allocation mechanism of IEEE specification. Then, we model the behavior of GTS allocation sing Markov chain. Details follow in the seqel. A. Nmber of Garanteed Time Slots In this sbsection, we derive the nmber of GTSs ( ) that can be allocated as a fnction of IEEE protocol parameters (BO, SO) and we derive the nmber of time-critical data packets τ n for each GTS reqest that can be served, and the imm forward delay D FS that a packet experiences to be transmitted in the GTS. According to the standard [1], it is possible to transmit more than one data frame inside one allocated GTS. The only restriction is that the transactions (inclding IFS and acknowledgement if reqired) mst complete before the end of the GTS. Assme that each device reqests to send the nmber of data frames τ n with the packet forward delay D FS inclding IFS and the transmission delay in symbol nit. The length of the packet forward delay is dependent on the size of the frame as follows D FS = D SIFS = D LIFS = L fr R s R b + macminsifsperiod if L fr amaxsifsframesize, + macminlifsperiod otherwise. (3) L fr R s R b where L fr is the frame length, R s is the symbol rate, R b is the bit rate. Let the dration of sperframe slot be T SS, then T SS = T SD N SF = T SFD SO 4, (4) where T SD is the nmber of symbols forming a sperframe, N SF is the nmber of slots contained in a sperframe, and T SFD is the nmber of symbols forming a sperframe when SO =. Consider that a single GTS may extend over a nmber of sperframe slots θ n. Since a given GTS needs to be greater than the total forward delay τ n D FS, it follows that τn D FS θ n θ min =, (5) T SS where θ min is the minimm nmber of sperframe slots for a single GTS to serve the data frames τ n D FS. By considering the minimm length of a CAP, T cap T min,cap, the constraint of the imm nmber of GTS that PANC can allocate in a sperframe is given by T SD T cfp = T SD T SD N SF k θ min T min,cap, (6) where k is the nmber of waiting reqests and T cfp is the nmber of symbols forming a CFP. By considering that the imm nmber of GTSs is 7 in any sperframe, it reslts that the imm nmber of GTSs to be allocated to devices is k = min N SF ( 1 Tmin,cap θ min T SD ), 7. (7) Note that is a fnction of IEEE parameters, e.g., (BO, SO), and application constraints, e.g., the nmber of data packets τ n and total imm forward delay D FS. In addition, a lower SO reslts on a decreasing of the imm nmber of GTSs. B. Markov Chain Model Here we develop a Markov model to stdy the behavior of the GTS allocation mechanism, see Fig. 3. Let t be a positive integer representing the time progress as expressed in sperframe nits. In other words, t is the sperframe conter. Note that t corresponds to network time de to the beacon enabled mode. The sperframe conter t = 1 correspond to the first sperframe of the network withot any waiting reqests at the PANC. Let i be the probability of i sccessfl reqests dring CAP, given that the imm nmber of reqests is L. If the arriving reqests observe that the qee size of already waiting reqests is over a threshold B, then these arriving reqests are dropped becase of the time limitation N GPT. Note that B is fixed and it is given by (N GPT + 1) where is the imm nmber of GTS for each sperframe and N GPT = agtsdescpersistencetime. After N GPT sperframes later, the GTS description of the beacon is removed, see details in [1]. When there are k reqests, k B, then some reqests k will be delayed to obtain GTS in the next sperframe. Let n(t) be a non-negative integer representing the nmber of waiting reqests at the beginning of the crrent sperframe t. Then n(t) {,, B, B}, where we denoted by B that some of the new arrived reqests are dropped de to the limited qee size B. If the reqests are over B after the new arrivals, then some of the new reqests will be dropped on the base of a FCFS fashion. We assme that there is no reallocation of GTS and each reqest is allocated at one GTS. By this assmption, the pair {t, n(t)} can be sed to bild the two-dimensional Markov chain reported in Fig. 3. The transition probabilities of the chain are described by the following eqations: P {t + 1, t, i} =, for i <, B, (8) P {t + 1, t, i} = + i, for i, i, B (9) P {t + 1, t, i} =, for i, < i, B, (1) P {t + 1, B t, i} = B, for i <, (11) P {t + 1, B t, i} = B + i, for i, (1) where t 1 and k = 1 k i= i = L i=k+1 i. Note that the imm nmber of reqests is L. Eq. (8) shows the probability that reqests arrive when there are less than the imm nmber of GTSs reqests at sperframe t. Note that GTSs will be allocated to previosly arrived reqests

4 1, 1 B 1 +1 B B,,1,, + 1 B B B B B 1 B 1,B 1,B,B , 3,1 B B 3, 3, + 1 3,B 1,B B B ,B t-1, B t-1,i B t 1, B i + t-1, B + B + t 1,B t 1,B i t, t,i t, t, B B B B + t,b t,b Fig. 3. Markov chain model for the GTS allocation of the CFP period. before CAP of the sperframe t i.e., all waiting reqests at the beginning of the sperframe t + 1 arrive dring CAP of the sperframe t. Eq. (9) gives the probability that reqests are remained at the sperframe t+1 when there are some reqests i at the sperframe t. The PANC allocates a imm nmber of GTS to serve the reqests i. Frthermore, since the nmber of waiting reqests is larger than, some reqests will fail to get GTS at the sperframe t. Hence, there are two different grops of reqests at the sperframe t+1, one grop of reqests arriving before the sperframe t and another grop of reqests arriving at the sperframe t. Let s call the first grop as old reqests and the second grop as new reqests throghot this paper. Eq. (1) shows the probability that there is less nmber of reqests in the sperframe t + 1 than i if there are i reqests at sperframe t. Note that the transition probability is eqal to de to the imm nmber of GTSs. Eqs. (11) and (1) show the transition probabilities of the drop state B for the reqests at sperframe t. Let s denote the probability that the process described by the Markov chain is in state k at time t as follows: π t k = P {t, k} for t {1,, } and k {,, B, B }. Using the transition probabilities given by Eqs. (8) (1), we derive the state probabilities at sperframe t + 1 as 1 + π t+1 = πi t + πi t + i, i= i= for + < B, B, (13) 1 π t+1 = πi t + πi t + i + πb t + B, i= i= for + B, B, (14) π t+1 B 1 = πi t B + πi t B + i + πb t, i= i= for = B, (15) where t {1,..., } and i, {,..., B, B }. The state probabilities of Eqs. (13) and (14) are given by the sm of two parts, de to the nmber of waiting reqests at the sperframe t. Denote by π = lim t π t R 1 (B +) the stationary distribtion and by P R (B +) (B +) the transition probability matrix [13]. For nmerical calclations, it is convenient to se the following expression: π = e (P I + E) 1, where I R (B+) (B+) is the identity matrix, E R (B +) (B +) is the matrix with all the elements eqal to 1, and e R 1 (B +) is the vector with all elements eqal to 1. From the Markov chain of Fig. 3, we see that the transition probability matrix is P = 1 +1 B B B B 1 B 1 B , (16) where k = 1 k i= i = L i=k+1 i.

5 IV. PERFORMANCE ANALYSIS OF GTS ALLOCATION In this section, we bild on the modelling of the GTS allocation developed in previos section by the Markov chain to analyze performance of GTS allocation in terms of stability (expected nmber of waiting and dropped reqests, and qee overflow probability), delay of serving the reqests, and throghpt. A. Stability Analysis Here we give the expected nmber of GTSs reqests by the devices waiting to be served, the expected nmber of GTSs reqests that are dropped becase of limited bandwidth, and the qee overflow probability. Using the state probability πk t of the Markov chain derived in the previos section, we can compte the mean nmber of waiting reqests of the PANC at the sperframe t by E [r(t)] = k= k π t k + B π t B. (17) Note that the nmber of reqests is related to the delay of GTS allocation de to a FCFS fashion for the qee management. From the Markov chain model, we see that the expected nmber of dropped reqests is given by E [r d (t)] = + 1 πi t i= k=b +1 i= k=b i π t B k= +1 k (k B ) π t i k (k (B i + )) k (k ). (18) Let Pover t+1 denote the qee overflow probability of the reqests in the sperframe t + 1. Then 1 Pover t+1 = 1 = i= i= + π t B πi t k=b +1 k= +1 π t i B + k + k i= k=b i+ +1 π t i k i= π t i B i+ + π t B. (19) Finally, to analyze the stability of GTS allocation, we can se the stationary distribtion given by Eq. (16) to derive the limit as t tends to infinity of the expected nmber of waiting reqests, the expected nmber of dropped reqests, and the qee overflow probability. B. Delay Analysis In this sbsection, we analyze the expected delay of GTS allocation, namely the average delay between the arrival of a new reqest for GTSs by a device for a time-critical packet and its effective allocation in some of the next sperframes. The PANC determines a device list for GTS allocation in the next sperframe based on a FCFS fashion. When new reqests are received in a CAP, then the delay of GTS allocation can be estimated by observing the qee size of waiting reqests. Note that PANC makes a preliminary decision whether it is able to serve a reqest or not. Assme that the arrival process of reqests is niformly distribted dring CAP. Hence, the mean delay between the arrival time in CAP and the end of CAP at a sperframe is half of CAP period. Sppose that reqests arrive at the sperframe t and observe i reqests already waiting in the qee. We can calclate the nmber of sccessfl reqests that get a GTS and delayed reqests that fail to obtain a GTS at the next sperframe, as explained in the following. Based on FCFS fashion, new reqests are able to obtain GTS after i old reqests arriving before the crrent sperframe t. We first compte the nmber of sperframe to allocate GTS for old reqests (see Fig. 4). Notice that i old reqests reqire to se q i + 1 sperframes, pls r i remainders that obtain the GTSs later, after q i + 1 sperframes have been served. Hence, i old reqests reqire q i + sperframes to obtain GTS if there are r i remainders. Note that r i remainders will share the CFP along with the new reqests at the sperframe t + q i + 1, as represented in Fig. 4. It follows that ( ) q i = i, r i = rem i, () where q i is an integer qotient and the fnction rem retrns an integer remainder. The nominator i of Eq. () takes into accont the crrent GTS allocation mechanism of sperframe t. Note that the new reqests are not able to get GTS at the crrent sperframe t since beacon incldes information of GTS allocation at each sperframe. By considering the waiting time of old reqests, it is possible to calclate the delay of new reqests. The qotient q i of old reqests can be considered as a delay offset in the comptation of the delay of the new reqests. Note that we consider r i remainders by smming it with new reqests de to the shared CFP with r i remained reqests (see Fig. 4). By a similar approach sed to derive Eq. (), the qotient and remainder of new reqests are given by ( ) q = +ri, r = rem +ri. (1) The new reqests reqire q + 1 sperframes to complete GTS allocation. By looking at Fig. 4, r waiting reqests will remain waiting to obtain GTS at the sperframe t+q i +q +1. To analyze the delay, we define the beacon interval as T t,t+1 = T t cfp + T t sp + T t+1 cap. () where Tcfp t is a CFP at sperframe t, T sp t is a inactive period at sperframe t, and Tcap t+1 is a CAP at sperframe t+1. Let D i,,t denote the expected delay of new reqests when there are i old reqests at the sperframe t. We distingish the nmber of sccessfl reqests that obtain GTS ot of new reqests for a different sperframe. Note that new reqests of the sperframe t will obtain GTS from the sperframe t + q i + 1 to the sperframe t + q i + q + 1. From Fig. 4, we see that the delay of allocating GTS is the sm of three components: Arrival delay of CAP: Since the arrival time of new

6 new reqests Sperframe t Sperframe t+1 q i Sperframe t+ +1 q i Sperframe t+ + q i Sperframe t+ +3 Sperframe t+ q i + q +1 ri ri r q i+1sperframes Fig. 4. Waiting time of new reqests that observe i old reqests at sperframe t. q Sperframes reqests is niformly distribted in CAP period, the arrival delay mst be based on FCFS fashion. Offset delay of old and new reqests : The new reqests need to wait a nmber of sperframes before obtaining GTS since there are old and new reqests arriving before the crrent reqest. Service delay of CFP: If the new reqests are able to obtain GTS at the crrent sperframe, then the sperframe slot size T SS and the minimm nmber of sperframe slots θ min need to be acconted for. Hence, it is possible to describe the delay of new reqests by considering the sperframe time to obtain GTS, as given by the following claim: Claim 1: The sperframe time delay to obtain GTS is d g (n) = t+qi + T t cap + θ min T t+q i+1 SS (r i + +1 for < r (q = ) ( ) t+qi + Tcap t 1 r i +θ min T t+q ( i+1 SS ri + r i ) +1 for r (q > ), n = t + q i t+qi + ( n l=t+q T t i+1 cap 1 ri (n (t+q i+1)) ) n+1 θmintss + ( +1) ) + T l,l+1 for r (q > ), t + q i + ( 1 n t + q i + q 1 1 ri (q 1) t+qi + t+q i+q l=t+q i Tcap t ) r + T l,l+1 + θ mint t+qi+q +1 SS (r +1) for r (q > ), n = t + q i + q. (3) where q i, q is the qotient and r i, r is the remainder of old and new reqests, respectively. Proof: If the new reqests are < r (i.e., q = ), then the average of arrival delay is half of T cap. Frthermore, the average of service delay is the half of the service time of + 1 new reqests with a single GTS dration θ min T t+q i+1 SS for each reqest. Note that r i remainders are considered as an offset before the GTS allocation of new reqests. The offset qi +t takes into accont the delay of i old reqests. If new reqests are more than r (i.e., q > ), then it is possible to categorize the delay into three grops (initial, intermediate, end delay). The different grop of delays are also combined with the arrival, offset, and service delay components. The initial delay is the delay of first allocated GTSs r i ot of new reqests at the sperframe t+q i +1, see Fig. 4. The average of arrival delay of the initial delay grop considers the ratio 1 r i since the arrival time of new reqests are niformly distribted dring CAP. With a similar approach of < r, the average of the service delay considers r i remainders and r i new reqests. The intermediate delay is the delay between the first and last allocated GTSs (q 1) ot of new reqests from the sperframe t + q i + to t + q i + q, see Fig. 4. Hence, the nmber of allocated GTSs is the imm nmber of GTSs ot of new reqests. The average of arrival delay of the intermediate delay grop considers the ratio 1 ri (n (t+q i +1)) since other reqests of the nmber r i of the initial delay grop and the nmber (n (t+q i +1)) of the intermediate delay grop obtain GTSs before new reqests ot of. Note that the nmber of allocated GTS is dependent on the time progress n (sperframe nit). With a similar approach sed in the case < r, we see that the average of the service delay considers new reqests. The end delay is the delay of last allocated GTSs r ot of new reqests at the sperframe t + q i + q + 1, see Fig. 4. The average of arrival delay of the end delay grop considers the ratio 1 ri (q 1) r since other reqests of the nmber r i of initial delay grop and the nmber (q 1) of intermediate delay grop obtain GTSs before r remainders ot of new reqests. With a similar approach of the case < r, we see that the average of the service delay considers r remainders. By considering Eq. (3), the expected delay D i,,t of new reqests that oberserve a qee size of i waiting reqests at sperframe t is t+qi + T t cap + θ min T t+qi+1 SS (r i + +1 ) D i,,t = for < r r i d g (t + q i ) + t+q i+q 1 n=t+q i+1 d g (n) (q ) + r d g(t + q i + q ) for r. (4) where (t) is the nit step fnction. Hence, the following claim holds: Claim : The expected delay experienced by new reqests arriving at sperframe t is 1 E[D(t)] = πi t L i= =1 k=1 k +D i,b,t ( (B + 1))} + i= =1 π t i {D i,,t (B ) L k=1 {D i,,t (B i + ) k

7 +D i,b i+,t ( (B i + + 1))} + π t B =1 L k=1 {D B,,t ( ) k +D B,,t ( ( + 1))}, (5) where D i,,t is the estimated delay by sing the Eq. (4) and L is the imm nmber of reqests. Proof: The first term of Eq. (5) presents the average delay of new reqests when the old reqests i 1. By considering the Markov chain, if there are more than B new reqests, then the extra reqests ot of B arrivals will be dropped. The nit fnction is sed to take into accont the limited qee size B. The second term considers the average delay when the old reqests i B. Note that the Markov chain represents the feasible nmber of new reqests withot the dropped reqests. With similar approach of first term, it is possible to consider the imm nmber of new reqests, B i+, withot dropped reqests when i old reqests wait in the qee. The third term comptes the average delay at the dropped state B, see the Markov chain. At the dropped state B, new reqests are considered since the qee size of old reqests is B. The average delay mainly depends on the traffic pattern i of the nmber of GTS reqests and protocol parameters (BO, SO) of, D i,,t. It is possible to consider the average delay constraint by sing the specific qee size B. C. Throghpt Here we characterize the GTS throghpt, namely, the average amont of packets that can be transmitted dring a GTS. Let P s (t) be the probability that a GTS allocation is sccessfl at the sperframe t. Then P s (t) = 1 P drop (t) (6) E [r d (t)] = 1 ( L k=1 B ), kk i= πi t + πt B where P drop (t) is the drop probability de to the limited qee size B at the sperframe t. Note that P drop (t) is the ratio between the mean nmber of dropped reqests, given by Eq. (18), and the mean nmber of total reqests at the sperframe t. As a nmber of reqests increase, it increases the length of waiting qee and reslts on higher dropped probability. If we assme that the frame size D FS is smaller than T cfp, then the normalized system throghpt S(t) of the sperframe t is given by the ratio of the average length of sccessflly allocated payload in a GTS time slot to the average length of a GTS time slot, namely S(t) = P S(t) L pl τ n θ min T SS, (7) where L pl is the length of payload of each data packet, τ n is the nmber of data packets, T SS is the length of sperframe slot, and θ min is the minimm nmber of sperframe slots, which are given by (4) and (5), respectively. The normalized system throghpt S(t) depends on the traffic pattern since the drop probability P drop (t) is related to the nmber of data packets τ n, the frame size L fr, the mean and variance of reqests. Hence, the system throghpt is related to the time effectively sed for data transmission within a GTS. It is possible to derive the optimal protocol parameters (BO, SO) which imize the throghpt of GTS sage. V. NUMERICAL RESULTS Here we present extensive Monte Carlo simlations of the GTS allocation to validate or theoretical reslts, which we then se for a performance analysis. The simlations are based on the specifications of the IEEE [1], with beacon order set eqal to sperframe order, namely BO=SO. In the simlations, the nmber of GTS reqests for each sperframe follows some different probability density fnctions. In particlar, the simlation experiments are obtained with Poisson, Normal, and Gamma distribtion. We considered the Gamma distribtion becase many other distribtions can be approximated by it. Frthermore, we investigated the effects of the protocol parameters (BO, SO) in terms of throghpt and delay. Details follows in the seqel. A. Validation We validated the average nmber of reqests, average delay of GTS allocation, the average nmber of dropped reqests and the qee overflow probability as the time progress. Recall that the average nmber of dropped reqests and the qee overflow probability are the important metrics for the stability of qee management. Figs. 5(a), 5(b) compare the average nmber of waiting reqests given by Eq. (17) and the average delay of reqests given by Eq. (5) with simlation reslts for different distribtions of the nmber of GTS reqests, respectively. The analytical model follows well the simlation reslts. By comparing Fig. 5(a) and Fig. 5(b), we confirm the strong dependence between the average nmber of reqests and the average delay de to a FCFS fashion of the qee management. As a nmber of GTS reqests increase dring the CAP, the length of waiting qee and average delay increase. Figs. 6(a), 6(b) report the average nmber of dropped reqests given by Eq. (18) and the overflow probability as obtained by Eq. (19) with simlations, respectively. Observe that or analytical model matches well the average overflow probability for every distribtion of the nmber of GTS reqests. Therefore, we conclde that we can apply the average nmber of dropped reqests and the overflow probability to analyze the stability of the qee management, as we see next. In addition, since a nmber of GTS reqests depend on the random access scheme of CAP, it will be interesting to investigate the impact of reliability in CAP to the stability isse of qee management of GTS allocation. B. Effect of Beacon and Sperframe Order In this section, we investigate the impact of beacon order on the average throghpt and delay when the Poisson distribtion is assmed for the nmber of GTS reqests. We remark here that similar behaviors as those investigated by sing the Poisson distribtion are observed by adopting the Normal and Gamma distribtions.

8 5.8 Average nmber of reqests Simlation reslts for Normal (mean=7, var = 1) Analytical model for Normal (mean=7, var = 1) Simlation reslts for Poisson (mean = 7) Analytical model for Poisson (mean = 7) Simlation reslts for Gamma (K = 1) Analytical model for Gamma (K = 1) time (in sperframes) Average nmber of dropped reqests Simlation reslts for Normal (mean=7, var = 1) Analytical model for Normal (mean=7, var = 1) Simlation reslts for Poisson (mean = 7) Analytical model for Poisson (mean = 7) Simlation reslts for Gamma (K = 1) Analytical model for Gamma (K = 1) time (in sperframes) (a) Average nmber of reqests (a) Average nmber of dropped reqests Average delay (in seconds) Simlation reslts for Normal (mean=7, var = 1) Analytical model for Normal (mean=7, var = 1) Simlation reslts for Poisson (mean = 7) Analytical model for Poisson (mean = 7) Simlation reslts for Gamma (K = 1) Analytical model for Gamma (K = 1) time (in sperframes) (b) Average delay of GTS allocation Fig. 5. Average nmber of waiting reqests (a) and average delay of reqests (b) as obtained by simlations and Eqs. (17) and (5), respectively. The length of the payload is L pl = 4 bytes, the nmber of data packets is τ n = 3, the beacon order is BO = 4, the mean nmber of reqests is 7, the variance of reqests is 1 for the Normal distribtion, the mean nmber of reqests is 7 for the Poisson distribtion, and the shape and scale of the Gamma distribtion is eqal to 1 and 7, respectively. We evalated the average throghpt of one GTS given in Eq. (7) for different beacon order vales, as a fnction of the mean of the Poisson distribtion, whereas the length of data payload L pl, and the nmber of data packets τ n are fixed. Fig. 7 shows the effects of beacon order on the throghpt as a fnction of the mean nmber of reqests of the Poisson distribtion, and for a fixed length of payload (L pl = 4 bytes), a fixed nmber of data packets (τ n = ). Observe that the GTS allocation mechanism presents a worse behavior in terms of throghpt for BO>3. Indeed, high beacon orders reslt on increased amont of wasted bandwidth of an allocated GTS. Frthermore, if BO 1 and the mean nmber of reqests is greater than 6, then a decreasing throghpt occrs de to the increasing dropped reqests. Note that an increasing nmber of reqests increases the length of waiting qee and reslts on higher dropped probability, see Eq. (6). In addtion, it will be very interesting extension to define the global throghpt by considering the data transmission both CAP and CFP. Fig. 8(a) illstrates the throghpt as achieved by the analytical model and simlations as a fnction of the payload length L pl. The good configration of the beacon order that imizes the throghpt is arond BO=, 3 for a given dty cycle. However, the imm throghpt vale depends on the length of payload. Note that sch a dependance is Overflow probability Simlation reslts for Normal (mean=7, var = 1) Analytical model for Normal (mean=7, var = 1) Simlation reslts for Poisson (mean = 7) Analytical model for Poisson (mean = 7) Simlation reslts for Gamma (K = 1) Analytical model for Gamma (K = 1) time (in sperframes) (b) Qee overflow probability Fig. 6. Stability of the qee management as obtained by simlations and analytical model of Eqs. (18) and (19). The length of payload is L pl = 4 bytes, the nmber of data packets is τ n = 3, the beacon order is BO= 4, the mean nmber of reqest is 7, the variance of reqests is 1 for the Normal distribtion, the mean nmber of reqests is 7 for the Poisson distribtion, and the shape and scale for the Gamma distribtion are 1 and 7, respectively. Throghpt Simlation reslts (mean = 4) Analytical model (mean = 4) Simlation reslts (mean = 5) Analytical model (mean = 5)) Simlation reslts (mean = 6) Analytical model (mean = 6) Simlation reslts (mean = 7) Analytical model (mean = 7) Beacon Order (BO) Fig. 7. Effect of beacon order on the throghpt of GTS allocation as a fnction of the mean of the Poisson distribtion (4, 5, 6, 7). The length of the payload is L pl = 4 bytes, and the nmber of data packets is τ n =. very limited for BO>1 de to wasted bandwidth. Fig. 8(b) shows the throghpt as obtained by the analytical model and simlations as a fnction of the nmber of data packets τ n. The analysis predicts qite well the simlation reslts. The best configration of the beacon order that imizes the throghpt is a fnction of the nmber of data packets. Fig. 9 shows the average delay of a GTS allocation, for a given length of payload and nmber of data packets as a fnction of the mean of the Poisson distribtion. It is clear that the average delay depends on the mean nmber of reqests.

9 Throghpt Simlation reslts (τ n = bytes) = bytes) Simlation reslts (τ n = 4 bytes) = 4 bytes) Simlation reslts (τ n = 6 bytes) = 6 bytes) Simlation reslts (τ n = 8 bytes) = 8 bytes) Beacon Order (BO) (a) Throghpt as a fnction of L pl (, 4, 6, 8 bytes), the nmber of data packets τ n =. Throghpt Simlation reslts (τ n = 1, L pl = 4 bytes) = 1, L pl = 4 bytes) Simlation reslts (τ n = 4 bytes) = 4 bytes) Simlation reslts (τ n = 3, L pl = 4 bytes) = 3, L pl = 4 bytes) Beacon Order (BO) (b) Throghpt as a fnction of τ n (1,, 3), the length of payload (L pl = 4 bytes). Fig. 8. Effect of the beacon order on the throghpt of GTS allocation. The mean nmber of reqests of Poisson distribtion is 7. Average delay (in seconds) Simlation reslts (mean=4) Analytical model (mean=4) Simlation reslts (mean=5) Analytical model (mean=5) Simlation reslts (mean=6) Analytical model (mean=6) Simlation reslts (mean=7) Analytical model (mean=7) Beacon Order (BO) Fig. 9. Effect of beacon order on the average delay of GTS allocation as a fnction of the mean of the Poisson distribtion for the nmber of GTS reqests (4, 5, 6, 7), for a fixed length of the payload (L pl = 4 bytes) and if the nmber of data packets τ n =. Notice that increasing the mean nmber of reqests increases also the mean delay becase of the longer waiting time, see Fig. 5. When the imm nmber of reqests is greater than 7, the average delay is monotonically increasing, i.e., we see an nstable qee size becase the imm nmber of GTSs per sperframe is 7. The monotonic relation between average delay and BO does not hold for BO =, 3 of the mean nmber of GTS reqests (5, 6). This behavior is explained by looking at the imm nmber of GTSs given by Eq. (7). Note that a lower BO reslts on a decreasing of the imm nmber of GTSs. By observing the traffic pattern of the nmber of GTS reqests, it is possible to optimize the protocol parameters (BO, SO) which garantee the average delay constraint. VI. CONCLUSIONS In this paper, we presented an analytical model based on a Markov chain to compte the performance of the GTS allocation mechanism in IEEE standard. Monte Carlo simlations validated the analysis. Or theoretical analysis gives accrate nmerical reslts, which are different from the ones obtained in [1] by sing the network calcls. We evalated the stability of the qee size at the network coordinator, the delay to serve a GTS reqest, and the achieved throghpt for different traffic patterns and protocol parameters. We derived the dependance of the average delay and qee size as a fnction of the nmber of reqests. Frthermore, we analyzed the achieved throghpt as a fnction of the amont of data packets to forward for each reqest. We observed that lower beacon order gives lower delay bt ensres a worse throghpt becase of the higher drop probability. By contrast, higher beacon order increases significantly the average delay and degrades the throghpt de to wasted bandwidth. 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