Analysis of group paging with pre-backoff

Abstract

Group paging can simultaneously activate hundreds of user equipments (UEs) using a single paging message. Upon receiving the group paging message, all UEs should immediately transmit their paging response messages through the random access channels (RACHs). Simultaneous channel access from a huge group of UEs may result in severe collisions of the RACHs during a very short period of time. In this paper, we propose a pre-backoff method to reduce the collision probability of random access requests. We develop an analytical model to investigate the performance and optimize the setting of the pre-backoff method. The accuracy of the analytical model is verified through computer simulation. Results show that the proposed pre-backoff method can effectively enhance the performance of group paging.

1 Introduction

Machine-type communication (MTC) is one of the most important services for next generation networks. 3GPP, IEEE and other international organizations adopt their technologies to the emerging market of the MTC [1]. Mass machine devices that simultaneously access the network may result in heavy collisions and thus, degrade the network performance [1,2]. In 3GPP [3], a concept of group paging has been introduced to alleviate the heavy collision problem in the random access channels (RACHs) of the cellular networks. However, no normative specification work on group issues took place. It may restart again in the next 3GPP Release 13. In group paging, the base station, which is known as an evolved-NodeB (eNB) in 3GPP, can assign a number of machine devices to form a paging group. The eNB can then simultaneously activate the group of machine devices by sending a single group paging message. Upon receiving the group paging message, all machine devices belonging to the paging group should immediately perform the random access (RA) procedure to access the network during a period of time, which is referred as a group paging interval herein [4].

The concept of grouping is also adopted by IEEE 802.11ah [1] to simplify operation with a huge number of associated machine devices. IEEE 802.11ah developed a restricted access window (RAW) mechanism to limit the set of machine devices accessing the channel and to spread their access attempts over a long period of time. In other words, RAW divides stations into groups, splits the channel into slots, and assigns each slot to a group. The machine devices can only transmit in the slot assigned to their group [1]. In IEEE 802.11ah, the base station, which is known as an access point (AP), may wake up a huge number of machine devices by setting the association identity bits in a traffic indication map (TIM) element carried by the beacon frame [1].

The related works of group paging are briefly summarized as below. The performance analysis of group paging via computer simulations was first addressed in [5]. The first analytical model of a simplified group paging without random backoff was first presented in [6]. The model is then completed and extended in [4] to analyze the performance of group paging by considering all the implementation constraints of the RA procedure specified in LTE networks. A group-based time control mechanism was presented in [7] to deal with radio access network overload problem in a femtocell-based MTC network. A priority grouping scheme is presented in [8] to deal with the RACH collisions by assigning priority-dependent backoff parameters. In [9], the authors proposed a two-level device partitioning scheme to deal with the RACH overload problem by limiting the number of contending machine devices. In [10], the authors proposed to alleviate the RACH collision problem by assigning dedicated preamble and RA slot for each UE. Our previous work in [2] introduces consecutive group paging, which provides several group paging intervals consecutively for a group. Normal group paging is performed in each interval, and failing UEs from the previous interval are paged again in the successive interval. Compared with the original group paging, this method slightly increases the success probability at the cost of increased access delay.

In group paging, heavy RACHs collisions are found at the beginning of the group paging interval due to the simultaneous channel accesses from the whole group of machine devices. A randomized paging response method has been proposed in 3GPP SA2 to deal with the heavy collision problem but without any follow-up work. In [11], the authors proposed a pre-backoff scheme to relief the heavy collision problem. The pre-backoff scheme enforces each machine device to perform a random backoff before its first preamble transmission. A preliminary study of the pre-backoff scheme in group paging via computer simulations was presented in [12]. The results showed that the pre-backoff scheme may properly reduce the heavy collision problem. However, it is not clear how to determine the optimum pre-backoff window size under a given group size.

This paper presents an analytical model to analyze the performance of pre-backoff scheme for group paging. The optimal setting for the pre-backoff window is then derived. The accuracy of the analytical model was then verified through computer simulations. Instead of performing paging consecutively, the method proposed in this paper spreads the first preamble transmission by using a pre-backoff mechanism to increase the success probability with lower access delay compared with the original group paging. The rest of the article is organized as follows. Section 3 summarizes the system model. The concept of group paging with pre-backoff is briefly introduced. The analytical model of group paging with pre-backoff is then elaborated. Simulation results are shown in Section 3. Section 3 draws the conclusion.

2 System model

2.1 Random access procedure in group paging

The random access model considered in the paper follows the LTE random access procedure standard. In LTE, time is divided into fix-length radio frames. Each radio frame consists of 20 sub-frames. One or more sub-frames in a radio frame are reserved for UEs to perform random access. These special sub-frames are referred to as RA slots herein. Each RA slot contains R RAOs. In group paging, a group paging interval is reserved for a group of M UEs. There are I _max RA slots in a group paging interval. Let $T_{\textit {RA\_REP}}$ be the number of sub-frames between successive RA slots. All symbols used herein are summarized in Table 1.

Table 1 System parameters and symbols used in the equations

The whole RA procedure assumed in this paper is shown in Figure 1. After being paged, all UEs randomly transmit one of the R preambles at the first RA slot (step 1). T _RAR is the processing time (unit: sub-frame) required by an eNB to detect the transmitted preambles. The eNB subsequently broadcasts random access response (RAR) messages to inform which UEs are acknowledged and granted for resource (step 2). W _RAR is the RAR window (unit: sub-frame), i.e., the time interval for the eNB to transmit RAR messages. Each RAR message can carry up to N _RAR acknowledgements. The maximum number of UEs that can be acknowledged during an RAR window, which is denoted by N _UL , is W _RAR N _RAR .

Figure 1

The steps in an RA procedure.

Preamble transmission may fail due to some conditions, i.e., not-detected (due to underpowered), collided (detected but not unique among other UEs which transmit preamble at the same RA slot) or not collided but not selected, since the eNB can only acknowledge N _UL preambles at a W _RAR . If an UE does not receive RAR in (T _RAR +W _RAR ) sub-frames after its preamble transmission, it performs backoff with the window of W _BO and then transmits the new preamble with an increased power at the immediate RA slot (step 3). UE can only attempt to transmit preamble up to N _PTmax times in one group paging interval.

After receiving RAR, the UE sends Msg3 at the next sub-frame (step 5). Each time the UE sends Msg3, it counts down T _CR sub-frames as a contention-resolution timer. The reason this timer is employed is to detect the collision of Msg3. In some cases, the eNB may successfully decode one preamble transmitted by multiple UEs because of constructive interference and then reply with a RAR accordingly, assuming that the preamble is transmitted by one UE instead of two. These two UEs will transmit their own Msg3 on the same resource and realize the random access failure after the T _CR expires without receiving anything from eNB.

The eNB sends NACK if it does not receive Msg3 in T _HARQ sub-frames after it sends RAR (step 6) or ACK otherwise (step 8). If UE receives NACK, it retransmits Msg3 in T _M3 sub-frames later (step 7). UE goes back to preamble transmission stage if Msg3 fails N _HARQ times. The eNB transmits Msg4 in $T_{A\_M4}$ sub-frames after it sends Msg3’s ACK (step 9). Subsequently, eNB waits for Msg4’s ACK in T _HARQ sub-frames. If Msg4’s ACK is not received, eNB retransmits Msg4 in T _M4 sub-frames later (step 10). The UE goes back to the preamble transmission stage if it still fails to receive Msg4 N _HARQ times (eNB can send Msg4 up to N _HARQ times). After receiving Msg4, the UE immediately send the Msg4’s ACK to finish the RA procedure (step 11).

2.2 Pre-backoff for group paging

This paper suggests the use of pre-backoff to enhance the performance of original group paging considered in [4]. In group paging with pre-backoff, each UE should perform pre-backoff for its first transmission and follow the standard LTE random access procedure for the re-transmissions. The pre-backoff timer is uniformly chosen in the range 0 to W _PBO .

The performance of group paging with pre-backoff during a group paging interval of I _max th RA slots is investigated. The group paging interval starts from the first RA slot and ends at the I _max th RA slot. The pre-backoff may postpone the transmission time of the first preamble up to the pre-backoff window, W _PBO . The group paging interval is given in (1).

$$ { \fontsize{8.3}{6}\begin{aligned} I_{{max}}=\left\lceil \frac{W_{{PBO}}}{T_{{RA\_REP}}}\right\rceil +1+(N_{{PTmax}}-1)\left\lceil \frac{T_{{RAR}}+W_{{RAR}}+W_{{BO}}}{T_{{RA\_REP}}}\right\rceil. \end{aligned}} $$

((1))

Let M _i,S [ n] and M _i,F [ n] be the number of successful and failed UEs that transmit the nth preambles at the ith RA slot, respectively; M _i [ n] be the total number of UEs that transmit the preamble in the ith RA slot; and P _P be the probability of an UE to correctly receive paging message. Upon receiving the paging message, P _P M UEs will immediately perform a uniform backoff before transmitting their first preamble. The total number of UEs that transmit their first preamble in the ith RA slot, M _i [ 1], is derived in (2).

$$ M_{i}[\!1]=\begin{cases} \frac{P_{P}M}{W_{{PBO}}} & \text{if}~~\, i=1,\\ \frac{T_{{RA\_REP}}P_{P}M}{W_{{PBO}}} & \text{if}~~\,2\leq i\leq\left\lfloor \frac{W_{{PBO}}}{T_{{RA\_REP}}}\right\rfloor,\\ \frac{(W_{{PBO}}\,\text{mod}\, T_{{RA\_REP}})P_{P}M}{W_{{PBO}}} & \text{if}~~ i=\left\lfloor \frac{W_{{PBO}}}{T_{{RA\_REP}}}\right\rfloor +1,\\ 0 & \text{otherwise}. \end{cases} $$

((2))

RA attempts in the first RA slot and the other RA slots are investigated separately, since there is no preamble retransmission in the first RA slot. The total number of successful UEs, M _1,S [ n], and failed UEs, M _1,F [ n], from the first RA slot is given in (3) and (4), respectively. Note that at the initial condition of i=1,M ₁[ n]=0 for n≠1.

$$ {\fontsize{7.8}{6} M_{1,S}[\!n]=\begin{cases} N_{UL} & \text{if}~~ n=1\,\text{and}~~ M_{i}[\!1]e^{-\frac{M_{i}[1]}{R}}p_{1}\geq N_{UL},\\ M_{i}[\!1]e^{-\frac{M_{i}[1]}{R}}p_{1} & \text{if}~~ n=1\,\text{and}~~ M_{i}[\!1]e^{-\frac{M_{i}[1]}{R}}p_{1}<N_{UL},\\ 0 & \text{if}~~ n\neq1; \end{cases}} $$

((3))

$$ {\fontsize{7}{6} M_{1,F}[\!n]=\!\!\begin{cases} M_{i}[\!1]-N_{UL} & \text{if} ~~n=1\,\text{and}\, ~~M_{i}[\!1]e^{-\frac{M_{i}[1]}{R}}p_{1}\geq N_{UL},\\ M_{i}[\!1]\left(1-e^{-\frac{M_{i}[1]}{R}}p_{1}\right) & \text{if} ~~n=1\,\text{and}\, ~~M_{i}[\!1]e^{-\frac{M_{i}[1]}{R}}p_{1}<N_{UL},\\ 0 & \text{if} ~~n\neq1. \end{cases}} $$

((4))

where N _UL is the maximal total number of UEs that can be acknowledged in one response window and p _n is the detection probability of the nth preamble transmission due to power-ramping effect [6], where p _n =1−(1/e ⁿ). Failed UEs in the first RA slot will perform backoff and retransmit again in the following RA slots. The total number of successful and failed UEs for each attempt from each RA slot, M _i,S [ n] and M _i,F [n], can be recursively obtained from (5) and (6), respectively.

$$ {\fontsize{8.44}{6}M_{i,S}[\!n]=\begin{cases} M_{i}[\!n]e^{-\frac{M_{i}}{R}}P_{n} & {\textstyle {\displaystyle \text{if}}}~~~\underset{{\scriptstyle n=1}}{\overset{{\scriptstyle N_{{PTmax}}}}{\sum}}M_{i}[\!n]e^{-\frac{M_{i}}{R}}p_{n}\leq N_{UL},\\ \frac{M_{i}[n]e^{-\frac{M_{i}}{R}}P_{n}}{\underset{{\scriptstyle n=1}}{\overset{{\scriptstyle N_{{PTmax}}}}{\sum}}M_{i}[n]e^{-\frac{M_{i}}{R}}p_{n}} & \text{otherwise}; \end{cases}} $$

((5))

$$ {\fontsize{7}{6}M_{i,F}[\!n]=\!\!\left\{\!\!\begin{array}{ll} M_{i}[\!n]\left(1-e^{-\frac{M_{i}}{R}}P_{n}\right) & {\textstyle {\displaystyle \text{if}}}~~\underset{{\scriptstyle n=1}}{\overset{{\scriptstyle N_{{PTmax}}}}{\sum}}M_{i}[\!n]e^{-\frac{M_{i}}{R}}p_{n}\leq N_{UL},\\ M_{i}[\!n]\left(1-\frac{P_{n}}{\underset{{\scriptstyle n=1}}{\overset{{\scriptstyle N_{{PTmax}}}}{\sum}}M_{i}[n]p_{n}}\right)N_{UL} & \mathrm{otherwise.} \end{array}\right.} $$

((6))

$$ M_{i}[\!n]\underset{{\scriptstyle k=K_{{min}}}}{\overset{{\scriptstyle K_{{max}}}}{\sum}}\alpha_{k,i}M_{k,F}[\!n-1]\qquad\text{if}\, ~~n>1. $$

((7))

The total number of UEs that conduct their nth RA attempt in the ith RA slot, M _i [ n], can be approximated by (7) for n>1. M _k,F [ n−1] in (7) represents the UEs that transmit their (n−1)th preambles at the kth RA slot but fail. α _k,i of these failed UEs conduct retransmission in the ith RA slot. K _min and K _max denote the minimal and maximal value of k, respectively.

The transition probability, α _k,i , its upper bound, K _max , and lower bound, K _min , in (7) can be derived based on the timing diagram given in Figure 2. α _k,i , K _max , and K _min are given in (8), (9), and (10), respectively. The UEs with failed preamble transmission at the kth RA slot may retransmit a new preamble at the ith RA slot only if the backoff interval of the kth RA slot is overlapped with the transmission interval of the ith RA slot. Therefore, α _k,i is the portion of the backoff interval of the kth RA slot which overlaps with the transmission interval of the ith RA slot (with k<i). In Figure 2, UEs which transmit their preambles at the kth RA slot at time $(k-1) T_{\textit {RA\_REP}}$ will recognize their RA failure after (T _RAR +W _RAR ) sub-frames. Each failed UE starts the backoff at time $(k-1) T_{\textit {RA\_REP}}+(T_{\textit {RAR}}+W_{\textit {RAR}})+1$. Therefore, the backoff interval of the kth RA slot starts from $(k-1)T+_{\textit {RA\_REP}}+(T_{\textit {RAR}}+W_{\textit {RAR}})+1$ and ends at $(k-1)T_{\textit {RA\_REP}}+(T_{\textit {RAR}}+W_{\textit {RAR}})+W_{\textit {BO}}$. The UEs transmit their preambles at ith RA slot if their backoff expires between the (i −1)th and the ith RA slot. Therefore, the transmission interval of the ith RA slot is $[\!(i-1)T_{\textit {RA\_REP}}+1 (i-1)T_{\textit {RA\_REP}}]$.

Figure 2

Timing diagram which shows the preamble retransmission.

K _min is obtained when the right boundary of the kth RA slot backoff interval reaches the left boundary of ith RA slot transmission interval, which is (K _min −1)T _{RA_REP} +T _RAR +W _RAR +W _BO ≥1+(i−2)T _{RA_REP} . Therefore, K _min is expressed as

$$ K_{{min}}=\left\lceil (i-1)+\frac{1-(T_{{RAR}}+W_{RAR}+W_{BO})}{T_{{RA\_REP}}}\right\rceil. $$

((8))

The maximal value of k (K _max ) is obtained when the left boundary of the kth RA slot backoff interval exceeds the right side boundary of the ith RA slot transmission interval, which is (K _max −1)T _{RA_REP} +T _RAR +W _RAR +1≤(i−2)T _{RA_REP} . Hence,

$$ K_{max}=\left\lfloor i-\frac{1+(T_{RAR} + W_{RAR})}{T_{RA\_REP}}\right\rfloor. $$

((9))

α _k,i can be determined based on k in the three cases shown in the lower part of Figure 2. In the first case, the right boundary of the backoff interval is within the transmission interval, i.e., $1+(i-2) T_{\textit {RA\_REP}} \le (k-1)T_{\textit {RA\_REP}}+T_{\textit {RAR}}+W_{\textit {RAR}}+W_{\textit {BO}} \le (i-1)T_{\textit {RA\_REP}}$. In this case, K _min ≤k≤i−((T _RAR +W _RAR +W _BO )/T _{RA_REP} ) and the overlapped region starts from the left boundary of the transmission interval and ends at the right boundary of the backoff interval. In the second case, the transmission interval is fully overlapped with the backoff interval. Therefore, the length of the overlapped region is $T_{\textit {RA\_REP}}$. In the third case, the left boundary of the backoff interval is within the transmission interval $1+(i-2)T_{\textit {RA\_REP}} \le (k-1)T_{\textit {RA\_REP}}+T_{\textit {RAR}}+W_{\textit {RAR}}+1 \le (i-1)T_{\textit {RA\_REP}}$. In this case, (i−1)−((T _RAR +W _RAR )/T _{RA_REP} )≤k≤K _max and the overlapped region starts from the left boundary of the backoff interval and ends at the right boundary of the transmission interval. α _k,i is the ratio of the overlapped region to the backoff interval and is expressed as

$$ {\fontsize{8.7}{6}\alpha_{k,i}=\begin{cases} \frac{(k-1)T_{RA\_REP} + T_{RAR}+W_{RAR}+W_{BO}-(i-2)T_{RA\_REP}}{W_{BO}}\\ \qquad\text{if}\, ~~K_{min}\leq k\leq i-\frac{T_{RAR}+W_{RAR}+W_{BO}}{T_{RA\_REP}},\\ \frac{T_{RA\_REP}}{W_{BO}}\\ \qquad\text{if}\, ~~i-\frac{T_{RAR}+W_{RAR}+W_{BO}}{T_{RA\_REP}}<k<i-1-\frac{T_{RAR}+W_{RAR}}{T_{RA\_REP}},\\ \frac{(i-1)T_{{RA\_REP}}-\left((k-1)T_{{RA\_REP}}+T_{{RAR}}+W_{{RAR}}\right)}{W_{{BO}}}\\ \qquad\text{if}\, ~~i-1-\frac{T_{{RAR}}+W_{{RAR}}}{T_{{RA\_REP}}}\leq k\leq K_{{max}},\\ 0\qquad\text{otherwise}. \end{cases}} $$

((10))

2.3 Performance metrics

The performance metrics are derived to estimate the collision probability, P _C , the successful access probability, P _S , and average access delay, $\overline {D_{A}}$. P _C is the ratio of the total number of collided preambles to the entire allocated preambles. P _S is the ratio of total UE that have successfully accessed the network to the total number of UE in the paged group. $\overline {D_{A}}$ is the total access delay for all successfully accessed UEs normalized by the total number of successfully accessed UEs. P _C ,P _S , and $\overline {D_{A}}$ are given in (11), (12), and (13), respectively.

$$ P_{C}=\frac{\underset{{\scriptstyle i=1}}{\overset{{\scriptstyle I_{{max}}}}{\sum}}\left(R-M_{i}e^{{-\frac{M_{i}}{R}}}-Re^{{-\frac{M_{i}}{R}}}\right)}{I_{{max}}R}, $$

((11))

$$ P_{S}=\frac{\underset{{\scriptstyle i=1}}{\overset{{\scriptstyle I_{{max}}}}{\sum}}\underset{{\scriptstyle n=1}}{\overset{{\scriptstyle N_{{PTmax}}}}{\sum}}M_{i,S}[\!n]}{M}, $$

((12))

$$ \overline{D_{A}}=\frac{\underset{{\scriptstyle i=1}}{\overset{{\scriptstyle I_{{max}}}}{\sum}}\underset{{\scriptstyle n=1}}{\overset{{\scriptstyle N_{{PTmax}}}}{\sum}}M_{i,S}[\!n]T_{i}}{\underset{{\scriptstyle i=1}}{\overset{{\scriptstyle I_{{max}}}}{\sum}}\underset{{\scriptstyle n=1}}{\overset{{\scriptstyle N_{{PTmax}}}}{\sum}}M_{i,S}[\!n]}, $$

((13))

where T _i is the access delay for the successfully accessed UEs from the transmitting preamble at the ith RA slot, which can be obtained from (14).

$$ T_{i}=(i-1)T_{{RA\_REP}}+T_{{RAR}}+W_{{RAR}}+\overline{T_{{MSG}}}. $$

((14))

In (14), $\overline {T_{{MSG}}}$ is the average time required by an UE to finish the message-part transmission. $\overline {T_{{MSG}}}$ is the time used to transmit u Msg3 and v Msg4 and is given in (15). Msg3 and Msg4 can be transmitted up to N _HARQ times. Msg3 is sent directly after the UE receives RAR message while Msg4 is sent $T_{A\_M4}$ sub-frames after eNB sends the Msg3’s ACK. Each of Msg3’s and Msg4’s transmission should be followed by ACK within T _HARQ sub-frames; otherwise, they should be retransmitted. Msg3’s retransmission is conducted after T _M3 sub-frames and Msg4’s is conducted after T _M4 sub-frames. Hence, $\overline {T_{{MSG}}}$ can be expressed as

$$ {\fontsize{8}{6}\overline{T_{{MSG}}}=\begin{array}[t]{l} \underset{{\scriptstyle u=1}}{\overset{{\scriptstyle N_{{HARQ}}}}{\sum}}\underset{{\scriptstyle v=1}}{\overset{{\scriptstyle N_{{HARQ}}}}{\sum}}P_{f}^{u+v-2}(1-P_{f})^{2}\times\\ \left((u-1)T_{{M3}}+(v-1)T_{{M4}}+(u+v)T_{{HARQ}}+T_{{A\_M4}}+1\right). \end{array}} $$

((15))

Note that higher P _S , lower P _C , or lower $\overline {D_{A}}$ are preferred.

Our proposed method of pre-backoff is used to spread the first preamble transmissions. An optimal value of W _PBO decreases P _C and thus increases P _S . However, further increase in W _PBO might decrease P _S since a too large W _PBO (relative to M) reserves longer time for the first RA attempt and reduces the opportunity for retransmission. Hence, the optimal W _PBO for a given value of M is the smallest value of W _PBO that yields the highest P _S . The optimal W _PBO here is effective for a delay-tolerant application since it might introduce an additional average access delay to reach the highest P _S . For an optimal value of W _PBO , the corresponding N _PTmax can then be determined based on the W _PBO using (1). In the implementation, the eNB can carry the optimal W _PBO and its corresponding N _PTmax along with the group paging message.

3 Results and discussion

The proposed analytical model is verified through computer-based simulation. Currently, there is no proper scenario that can be used to compare the performance of push-based and pull-based RAN overload control schemes. Hence, we did not compare group paging with the other push-based schemes. The analytical results are shown in line while the numerical results are shown in point, with each point representing the average value of 10⁷ samples.

Five values of W _PBO were considered in the simulations of Figures 3, 4, 5 and 6. The N _PTmax is adjusted for each W _PBO according to (1) to ensure common I _max for a fair comparison. Table 2 shows the setting of W _PBO and N _PTmax . In Figures 3, 4 and 5, Cases I and II, and P _P of 100% and 90% are examined. In Figure 6, Cases II to V and P _P of 100% are examined. The remaining parameters are I _max =55,R=54,N _UL =15,T _{RA_REP} =5,W _RAR =5,N _HARQ =5,T _RAR =2,T _HARQ =4,T _M3=4,T _{A_M4}=1, and T _M4=1.

Figure 3

Collision probability, P _C .

Figure 4

Access success probability, P _S .

Figure 5

Average access delay, $\overline {D_{A}}$ .

Figure 6

Collision probability, P _C .

Table 2 Parameter setting of W _PBO and N _PTmax

A group size of 10 to 1,000 (i.e., 0≤M≤1,000) is considered during the simulation. Figures 3, 4, and 5 depict P _C , P _S , and $\overline {D_{A}}$, respectively. Overall, the results show that the analytical model can accurately estimate the performance of group paging with pre-backoff under different P _P . In Figure 3, the cases with W _PBO ≠0 always have lower P _C and the case with the largest W _PBO yields the lowest P _C . It shows that the collision probability of group paging with pre-backoff always decreases as W _PBO increases. It is because pre-backoff spreads the UEs’ first RA attempts to multiple RA slots.

Figure 4 shows the access success probability. It is found that the access success probability does not increase as P _C decreases when W _PBO increases. This is because a larger W _PBO implies a smaller N _PTmax under the assumption of a fixed I _max . From the analytical results of Figure 4, P _S starts to fall below 1 at a smaller value of M in a larger W _PBO . P _S =1 can be maintained until M=364, 358, and 92 for W _PBO of 0, 30, and 150, respectively. Increasing W _PBO may decrease P _S for a small M because N _PTmax is decreased. However, the increase in W _PBO may increase P _S for large M since a larger pre-backoff window significantly reduces collision. Figure 4 shows that P _S =69% can still be achieved for 1,000 UEs with W _PBO =150. In larger M, smaller P _P increases P _S since it limits the contending UEs to an RA slot.

Figure 5 shows the average access delay, $\overline {D_{A}}$. It can be seen from Figure 5 that group paging without pre-backoff is suitable for a group size less than 160 UEs, since P _S =100% can be attained with the minimum average access delay. Pre-backoff with W _PBO =30 becomes useful when 160<M<357. In this region, the group paging with pre-backoff can attain P _S =100% with a lower average access delay. It is also found in Figure 5 that the average access delay is almost constant and is less dependent on M for the cases of W _PBO =150. It is because the pre-backoff delay dominates the average access delay. It is also found that group paging with pre-backoff can significantly increase the access success probability while keeping a lower average access delay for the successful UEs when we have a big group size (e.g., M=1,000).

As indicated in Figures 3, 4, and 5, an effective value of W _PBO is required to decrease P _C within acceptable $\overline {D_{A}}$. Figure 6 demonstrates the tradeoff between W _PBO and W _BO under the same constraint of N _PTmax =5. It is found that the minimum P _C is achieved when W _PBO =150 and W _BO =21. Among the cases, the highest value of W _PBO is 150 whereas the highest value of W _BO is 51. We did not include the case where W _BO =150 (equal to the highest W _PBO ) because it will have non-integer and too small value of N _PTmax to maintain the I _max =55. Too-small value of N _PTmax limits the RA attempt for each node, which will increase the total number of failed UEs. If higher W _BO is implemented without the smaller N _PTmax , the group paging interval might become too long, which means a large value of I _max . Therefore, increasing W _PBO is more practical to decrease collision compared with increasing W _BO .

Figure 7 shows the optimal values of W _PBO as a function of M for I _max =55. The optimal values are the smallest value of W _PBO that yields the highest P _S with the lowest $\overline {D_{A}}$ for a given value of M. Figure 7 depicts the relation between parameter settings and different scenarios of group size implementation. It suggests a value of W _PBO to be used for a certain group size. After a value of W _PBO is chosen, the setting of N _PTmax can be defined according to (1) for I _max =55.

Figure 7

Optimal W _PBO for each M to reach the highest P _S under I _max =55.

The simulation result shows that the proposed algorithm outperforms existing group paging when the group size is large. However, it is worthwhile to note that the result was obtained under an extreme assumption in which all the 54 preambles were reserved for group paging. In reality, the preambles are shared by all services and the network may only reserve five preambles for group paging. In this case, the proposed algorithm is expected to outperform the existing group paging when the group size exceeds 100.

4 Conclusions

In this paper, we suggested enhancing the performance of group paging by enforcing UEs to perform pre-backoff before the first transmission of preamble. The application had to be delay tolerant, since the time for backoff window might be long. An analytical model was then presented to derive the performance of group paging with pre-backoff. Simulations were conducted to evaluate the group paging with pre-backoff in terms of collision probability, access success probability, the average access delay, and the tradeoffs with the existing backoff mechanism. The results verify the accuracy of the analysis and demonstrate that pre-backoff can greatly enhance the performance of group paging when the group size is big. The proposed analytical model can be used to determine the optimal pre-backoff window size under a target access success probability.