Multi-Group Random Access Resource Allocation for M2M Devices in Multicell Systems Taesoo Kwon, Member, IEEE, and Ji-Woong Choi, Senior Member, IEEE

Index Terms—Machine-to-machine (M2M), stochastic geometry, random access, resource allocation.

400 Base station MT of group 1 MT of group 2 MT of group 3 Cell boundary



y−axis (meters)

Abstract—The efficient service of a tremendous number of machine-type devices with heterogeneous traffic types may consider simple random access as a solution. For more rigorous analysis of system performance in multicell environments, it is required to consider not only the packet collisions within the same cell but also the interference from other cells. This letter analyzes signal-to-interference-ratio distributions and derives efficient resource allocation schemes for spatial multi-group random access in multicell systems, using the Poisson point process model.





I. I NTRODUCTION F LATE, wireless communication industries have begun to discuss their scenarios serving machine-type communication devices such as meters/sensors as well as user equipments such as smart phones [1]. Such machine-type communication devices need intermittent uplink resources to report measured or sensed data to their serving base station (BS). It is however hard to dedicate limited uplink resources to each because of a huge number of them. Furthermore, machine-tomachine (M2M) traffics are quite heterogeneous; for example, smart metering has periodic traffics with relatively-loose delay requirements while e-health and vehicular communications have intermittent traffics with tight delay requirements. In order to provide a tremendous number of machine-type devices with these heterogeneous traffic types, simple random access can be considered as a efficient solution [2]. The main design issue for such methods becomes how to allocate resources for supporting heterogeneous requirements. For more rigorous analysis of system performance, it is required to consider not only the packet collisions due to simultaneous transmissions of machine-type (MT) devices within the same cell but also the interference from MT’s in other cells. Stochastic geometry provides a useful mathematical tool to model those geometric interferences [3]. Using this tool, the spectrum-sharing performance of multiple systems was analyzed in ad-hoc networks [4]. This letter analyzes signalto-interference-ratio (SIR) distributions for both underlay and overlay methods in multicell environments, and derives the efficient resource allocation methods that minimize the amount


Manuscript received December 27, 2012. The associate editor coordinating the review of this letter and approving it for publication was H. Wymeersch. This work was supported in part by MEST & DGIST (11-BD-0404). T. Kwon is with the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada (email: [email protected]). J.-W. Choi (corresponding author) is with the Department of Information & Communication Engineering, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu, 771-873, South Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2012.041112.112568


−400 −400



−100 0 100 x−axis (meters)




Fig. 1. Base stations and multi-group machine-type devices distributed as homogeneous Poisson point processes.

of resources required to meet outage-probability conditions of multiple groups, using the Poisson point process (PPP) model. II. S PATIAL R ANDOM ACCESS S UPPORTING M ULTIPLE G ROUPS An MT transmits its data to the closest BS. The MT’s are classified into Q groups according to their QoS requirements of which parameter may be a target outage probability considering a minimal SIR value required for successful receptions. The active MT’s of group q are distributed according to a homogeneous PPP, ΦM,q with intensity λM,q . The BS’s are also randomly deployed according to a PPP, ΦB with intensity λB , and use the same frequency. This environment provides a lower bound of the performance of systems with regularlydeployed BS’s [5]. Each active MT transmits its data to a BS nearest to it through a random access scheme, thus each BS builds a coverage based on Voronoi tessellation, as shown in Fig. 1. The standard power-loss-propagation model with the path-loss exponent α(> 2) and the Nakagami-m fading model are considered. Thus, the complementary cumulative distribution function (CCDF) of fading gain, G, can be given k m−1 by Pr {G > g} = exp(−mg) k=0 (mg) k! . It is assumed that Nakagami fading parameters of the desired signals and the interfering signals are ms and mi that are positive integers, respectively. This letter considers both underlay and overlay methods for sharing random-access-resources among groups [4], as shown in Fig. 2. Each cell served by a BS has total M resources

c 2012 IEEE 1089-7798/12$31.00 


Transmit Power

M Resources Group Q



Group 2


Group 1


Transmit Power

[Underlay method]

[Overlay method] MQ1 MQ 2

Group 1

Group 2


... Group Q


When their desired and interfering links respectively experience Nakagami-m fading with ms and mi that are positive ¯(u) , is defined integers, and the effective intensity of group q, λ M,q Q 2 as i=1 (Pi /Pq ) α λM,i /Mu , the CCDF of SIR of q-th group in the underlay method using Mu resources is given by   Pr SIR(u) >β = q

Time or Frequency

Time or Frequency


ms −1 k=0

1 k!




¯ λB +λ C(mi ,α)(ms β) α M,q (u)

l+k Δk,l l=0 (−1)

Fig. 2. Underlay and overlay methods for spatial multi-group random access.


2 ¯ (u) C(mi ,α)(ms β) α λ M,q

l 2

¯ λB +λ C(mi ,α)(ms β) α M,q (u)


for random access. The underlay method means that MT’s in all groups share all of M resources and MT’s in group q send data with transmit power Pq using one resource that is randomly chosen among M resources. MT’s are interfered from MT’s of all groups, and the transmission of each group is differentiated from other groups by controlling transmit power, Pq . On the other hand, The overlay method means that MT’s in group q share exclusively only a part of M resources, M νq (0 ≤ νq ≤ 1), and MT’s in group q send data with transmit power P using one resource that is randomly chosen among M νq resources. MT’s are interfered from only MT’s of the same group, and the portion of resources, νq , available for each group is controlled to differentiate it from others. A. SIR Distribution In dense networks, a main factor to degrade system performances is interference and this letter assumes that noise power can be neglected. The underlay method is first considered. A typical BS located on the origin receives the signal with Pq r−α GS from a typical MT of group q when the distance between them is r and the fading channel gain is GS . By Slyvnyak’s theorem [6], interfering MT’s of group i except for (q) a typical MT located on X0 still constitute a homogeneous PPP with intensity λM,i . When a cell has total Mu resources and an MT of group i transmits data using one chosen randomly among Mu resources, each resource serves 1/Mu of total MT’s in group i, on average. In other words, random selection of resources by MT’s of group i results in the thinning of ΦM,i with the retention probability 1/Mu and ˜ (u) [7]. If it yields a PPP with the intensity of λM,i /Mu , Φ M,i the interference is dealt with as noise and single antenna is of a equipped on both transmitters and receivers, the SIR(u) q link in group q is given by  (q) −α Pq X0  GS (u)  (i) −α (i) = Q  SIRq (1)   GI,j P i Xj (i) ˜ (u) i=1


X j



where GI,j denotes the fading gain of a link between a interfering node MT j of i-th group and a typical BS, and (i) (i) Xj is the location of MT j belonging to group i. {GI,j } are independently and identically distributed (i.i.d.) random variables. Lemma 2.1: Let MT’s of Q groups be distributed as PPP’s with intensity λM,q for q = 1, · · · , Q, respectively, and served by the nearest BS’s distributed as a PPP with intensity λB .

2 m− α Γ(1− α )Γ(m+ α2 ) where C(m, α) = Γ(m)  k−1 2 l j l j=0 (−1) j i=0 α (l − j) − i for l ≤

and Δk,l

(2) =

k. Here, Δ0,0 is defined as 1. Proof: See Appendix A. On the other hand, the SIR(o) of a link in group q in the q overlay method using Mo resources is given by  (q) −α Pq X0  GS (o)  (q) −α (q) , = SIRq (3)   GI,j (q) ˜ (o) Pi Xj X j



˜ (o) Φ M,q


where is a PPP with the intensity of MM,q , because MT’s o νq are interfered from only MT’s of the same group. Therefore, the SIR distribution of group q in the overlay method can be ¯ (o) = λM,q in (2). ¯(u) with λ directly obtained by replacing λ M,q M,q Mo νq The Mo is the number of resources for the overlay random access method. B. Resource Allocation To begin with, this subsection derives the minimum amount of resources under the Rayleigh fading environment (ms = mi = 1) to meet target outage probability, εt,q , for given minimum required SIR, βt,q , where q = {1, · · · , Q}, when there coexist MT’s of Q groups. Let Mu∗ and Mo∗ denote the minimum values of Mu and Mo , respectively. In order to meet an εt,q for a given βt,q in the underlay method when ms = mi = 1, from (2),   λB Pr SIR(u) > β ≥ 1 − εt,q (4) = 2 t,q q α ¯ (u) λB +λM,q C(1,α)βt,q

  2π where C(1, α) = Γ 1 − α2 Γ 1 + α2 = α sin(2π/α) . By the (u) ¯ definition of λM,q in Lemma 2.1 and (4), the conditions of Mu to meet εt,q ’s are given by  

 α2  2 C(1,α) Q Pi 1 α λM,i εt,q − 1 βt,q , ∀q . Mu ≥ λB i=1 Pq (5) Similarly, in the overlay method   λB Pr SIR(o) > β t,q = q (o)


α ¯ λB +λ C(1,α)βt,q M,q

≥ 1 − εt,q . (6)

¯ Thus, from (6) and the definition of λ M,q , the condition of Mo νq in the overlay method is given by

 2 1 α . (7) Mo νq ≥ C(1,α) λB λM,q εt,q − 1 βt,q (o)



Theorem 2.1: Given εt,q ’s and βt,q ’s where q ∈ {1, · · · , Q}, the minimum of Mo is given by 

 2 Q 1 α (8) λ − 1 βt,i Mo∗ = C(1,α) M,i i=1 λB εt,i

1 Group 1 @ Underlay P1=1.0000P1 0.9



Also, when Pq Pw



Group 1 @ Overlay 1=0.5870 Group 2 @ Overlay 2=0.2609 Group 3 @ Overlay 3=0.1522


α λM,i (1/εt,i −1)βt,i



0.5 0.4

1/εt,q −1 1/εt,w −1


βt,q βt,w ,

∀q, w ∈ {1, · · · , Q} ,



the minimum of Mu , i.e.  Mu∗ , is equal  to Mo∗ . Q Q Proof: From Mo = i=1 Mo νi , i=1 νi = 1, and (7), (8) and (9) can directly be obtained. Also, because the conditions in (5) can be reexpressed as   2 Q α · P λ Mu ≥ C(1,α) M,i i i=1 λB

 2 −2 1 α maxq Pq α εt,q − 1 βt,q (11) and Mu can be minimized by controlling the transmit power values, Mu∗ is derived as follows.   2 Q α Mu∗ = minP1 ,··· ,PQ C(1,α) i=1 Pi λM,i · λB

 2 −2 1 α maxq Pq α εt,q − 1 βt,q  2 Q α ≥ minP1 ,··· ,PQ C(1,α) i=1 Pi λM,i · λB  2

2 − 1 α Pi α εt,i − 1 βt,i 

 2 Q 1 α = C(1,α) i=1 λM,i εt,i − 1 βt,i λB = Mo∗

(12) 2 −2 1 α In (12), if (10) is met, i.e. values of Pi α εt,i − 1 βt,i for all i are the same, the equality holds. Under the Nakagami-m fading environment for general ms and mi that are positive integers, it is not easy to derive the minimum values of Mu and Mo in a closed form. However, a similar approach for efficient resource allocation can be (u) (o) λB applied. Let tq and tq be defined as 2 ¯ (u)





0.6 Probability

νq =  Q


Group 3 @ Underlay P =0.0672P 0.8

and, for this Mo∗ , 2 α λM,q (1/εt,q −1)βt,q

Group 2 @ Underlay P =0.1975P


λB +λM,q C(mi ,α)(ms β) α


¯ λB +λ C(mi ,α)(ms β) α M,q (o)

, respectively. In the underlay (u)

method, the CCDF of SIR in (2) is a function of tq whose (u) (u) domain is {tq |0 ≤ tq < 1} because C(m, α) > 0 for (u) (u) m ≥ 1 and α > 2. When f (tq ) denotes this function, f (tq ) is a monotonically increasing function over its domain with (u) f (0) = 0 and limt(u) →1 f (tq ) = 1. Its proof is omitted q because of space limitation. In fact, the increase in λB or the (u) number of resources, i.e. the increase in tq , always improves SIR performance. Therefore, the solution of an inequality (u) (u) f (tq ) ≥ 1 − εt,q can be expressed as tq ≥ τ (εt,q ) where (u) τ (εt,q ) is the solution of f (tq ) = 1 − εt,q . By replacing 1 − εt,q with τ (εt,q ) in (4), the condition of Mu under the Nakagami-m fading environment is obtained. This approach can also be applied to the overlay method where the CCDF (o) of SIR is expressed as f (tq ). Eventually, by replacing εt,q with 1 − τ (εt,q ) for all q in Theorem 2.1, similar results for the minimum values of Mo and Mu and the condition of Mo∗ = Mu∗ can be derived.

0.1 0 −40





10 SIR (dB)






Fig. 3. CDF of SIR for three groups in the underlay and overlay methods (lines: simulation results, symbols: analysis results; solid lines with symbols are for ms = mi = 1 while dotted lines with symbols are for ms = mi = 10).


Number of required resources

30 =3 =4 =5




15 (0.8449,0.1551) (0.7310,0.2690) (0.8357,0.1643)


(0.7077,0.2923) (0.6932,0.3068)









P /P 2 1

Fig. 4. Number of required resources vs. transmit power ratio in the underlay method when (βt,1 , βt,2 ) = (3dB, 0dB) (thin lines: (εt,1 , εt,2 ) = (0.1, 0.3), thick lines: (εt,1 , εt,2 ) = (0.2, 0.3); the solid circles and the numbers in brackets show Mo∗ and (ν1 , ν2 ) in the overlay method).

III. N UMERICAL R ESULTS AND C ONCLUSIONS This section evaluates and discusses the performance of random access schemes to support multiple groups. Fig. 3 shows the cumulative distribution function (CDF) of SIR for the underlay and overlay methods when each group  has an equal 3 intensity of MT’s, λB Mu = λB Mo = 8.0285 q=1 λM,q 1 , 3 and α = 4. Here, the values of λB Mu / q=1 λM,q = 8.0285, 3 λB Mo / q=1 λM,q = 8.0285, νq , and Pq were chosen based on the results in Theorem 2.1, in order for the resultant outage probability values to be (εt,1 , εt,2 , εt,3 ) = (0.1, 0.2, 0.3) when βt,1 = βt,2 = βt,3 = 0 dB. In other words, the allocations of wireless resources are designed to minimize the amount of resources required for the resultant outage probabilities of 1 This means that BS intensity is equal to 8.0285/M or 8.0285/M times u o total MT intensity.


(εt,1 , εt,2 , εt,3 ) = (0.1, 0.2, 0.3) and it is observed that the outage probabilities of each group under the Rayleigh fading (ms = mi = 1) are 0.1, 0.2, and 0.3 at the SIR of 0 dB, respectively, in Fig. 3. These νq and Pq values lead to the same CDF’s of SIR for underlay and overlay methods. In fact, the underlay multi-group random-access scheme have the same SIR performance as the overlay scheme in multicell systems when they allocate optimally its transmit power or resource portion to each group, in order to minimize the amount of resources required to meet given target outage probabilities of multiple groups. This figure also indicates that analysis results in Section II definitely coincide with the simulation results. Moreover, Fig. 3 shows that the line-of-sight factors of fading channels improve the SIR distributions. Fig. 4 shows the number of required resources according to transmit power ratio in the underlay method experiencing Rayleigh fading. It is assumed Qthat the BS intensity normalized by the MT intensity (λB / q=1 λM,q ) is one. In Fig. 4, the solid circles represent Mo∗ ’s and the numbers within brackets show νq ’s in the overlay method, which are respectively determined by (8) and (9). These results show that Mu∗ and Mo∗ are equal when Pq ’s are optimally controlled to minimize the amount of resources required for given outage probabilities. The path-loss exponent is another factor to affect the design of Mu or Mo and it means that the larger path-loss-exponent value leads to the less number of required resources because the effect of interference is reduced. From (2), it is observed that the SIR distributions of both underlay and overlay methods are closely related to the BS intensity. Thus, the geometric intensity of BS’s as well as conventional resources such as transmit power, time, and frequency, have to be considered for efficient resource allocations. For example, under Rayleigh fading environments, the number of required resources can be interpreted as λB Mu or λB Mo from (5) and (7). In fact, from (2)2 , it can be shown the BS intensity, λB , have the same effect on the SIR distribution as the amount of resources shared for random access, Mu or Mo νq , under environments that this letter assumes. Therefore, for improving outage probability, the bandwidth or the BS intensity should be increased. Even though random deployments of BS’s provide just a lower bound of their planned deployments, the analysis of this letter shows that the BS intensity is as important as conventional resources. These results will be able to be used as basic models for more sophisticated spatial resource managements. A PPENDIX A P ROOF OF L EMMA 2.1 This proof is similar to them in [4] and [8], but [4] considers only Rayleigh fading and [8] does not deal with multiple groups. For those differences and the completeness, this section briefly provides its derivation. Because BS’s and MT’s are distributed as PPP’s, the distance between a typical MT and its BS is Rayleighdistributed and its probability density function, fr (r), is equal 2 As mentioned in Section II-A, the SIR distribution of the overlay method ¯ (o) in (2). ¯ (u) with λ is expressed by replacing λ M,q M,q


to 2πλB r · exp(−λB πr2 ) [5]. Thus, the CCDF of SIR of q-th group is Pr{SIRq > β}   ∞  Pq r−α GS = 0 Pr > β fr (r)dr    ∞ Ir,q −1 = 2πλB 0 Pr GS > Pq βrα Ir,q r exp(−λB πr2 )dr (13) Q  (i) −α (i) where Ir,q = P |X | G . By the (i) j I,j i=1 Xj ∈ΦM,i i CCDF of Nakagami-m fading gain,   Pr GS > Pq−1 βr α Ir,q  k   ms −1 1 = k=0 k! (ms Pq−1 βr α )k EIr,q Ir,q exp  −ms Pq−1 βr α Ir,q (a) ms −1 1  dk = (−ms Pq−1 βr α )k dζ k LIr,q (ζ) k! k=0 −1 α ζ=ms Pq



where (a) follows from the definition of Laplace transform, LX (ζ) = EX {exp(−ζX)} and its property of Ltk X(t) (ζ) = (−1)k d


LX (ζ) . dζ k

The Laplace transform of Ir,q is

LIr,q (ζ) = EIr,q {exp(−ζIr,q )}


(i) (i) = i=1 EΦ˜ (u) { X (i) ∈Φ˜ (u) EGI,j {exp(−ζPi |Xj |−α GI,j )}} (b)







exp(−2πλ /Mu i=1  ∞M,q ∞ [1 − exp(−ζPi v −α g)]vdvfGI (g)dg) 0 0


= exp(−[




Piα λM,q /Mu ]πζ α C(mi , α))


where (b) follows from the probability generating functional (PGFL) of the PPP [6], fGI is the CDF of GI , and (c) follows from the change of variable v −α → x and the definition of the Gamma function. In order to evaluate the k-th order derivative of LIr,q (ζ), the following formula can be used. ∂k ∂z k

exp (f (z)) =  k l k (z)l−j exp (f (z)) l=0 l!1 j=0 (−1)j jl f (z)j ∂ f∂z k


From (13), (15), and (16), (2) is finally derived. R EFERENCES [1] S.-Y. Lien, K.-C. Chen, and Y. Lin, “Toward ubiquitous massive accesses in 3GPP machine-to-machine communications,” IEEE Commun. Mag., vol. 49, no. 4, pp. 66–74, Apr. 2011. [2] IEEE C802.16p-11/0034r1, “Group-based allocation of random access resources,” June 2011. [3] J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber, “A primer on spatial modeling and analysis in wireless networks,” IEEE Commun. Mag., vol. 48, no. 11, pp. 156–163, Nov. 2010. [4] J. Lee, J. G. Andrews, and D. Hong, “Spectrum-sharing transmission capacity,” IEEE Trans. Wireless Commun., vol. 10, no. 9, pp. 3053–3063, Sep. 2011. [5] J. G. Andrews, F. Baccelli, and R. K. Ganti, “Spectrum-sharing transmission capacity,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3122–3134, Nov. 2011. [6] D. Stoyan, W. Kendall, and J. Mecke, Stochastic Geometry and its Applications, 2nd edition. John Wiley and Sons, 1996. [7] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless Networks. NOW: Foundations and Trends in Networking, 2010. [8] T. Kwon and J. M. Cioffi, “Random deployment of data collectors for serving randomly-located sensors,” preprint available at http://arxiv.org/abs/1106.0840.

Multi-Group Random Access Resource Allocation ... - Semantic Scholar

J.-W. Choi (corresponding author) is with the Department of Informa- tion & Communication Engineering, Daegu ..... SIR (dB). Probability. Group 1 @ Underlay P1=1.0000P1. Group 2 @ Underlay P2=0.1975P1. Group 3 @ Underlay P3=0.0672P1. Group 1 @ Overlay 1=0.5870. Group 2 @ Overlay 2=0.2609. Group 3 ...

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