Diversity Polynomials for the Analysis ofTemporal Correlations in Wireless Networks

# Diversity Polynomials for the Analysis of Temporal Correlations in Wireless Networks

Martin Haenggi and Roxana Smarandache M. Haenggi and R. Smarandache are with the University of Notre Dame, Notre Dame, IN, USA. M. Haenggi is the corresponding author, his e-mail address is mhaenggi@nd.edu. This work has been supported by the NSF (grants CNS 1016742, CCF 1216407, and CCF 1252788). Manuscript date: July 3, 2019.
###### Abstract

The interference in wireless networks is temporally correlated, since the node or user locations are correlated over time and the interfering transmitters are a subset of these nodes. For a wireless network where (potential) interferers form a Poisson point process and use ALOHA for channel access, we calculate the joint success and outage probabilities of transmissions over a reference link. The results are based on the diversity polynomial, which captures the temporal interference correlation. The joint outage probability is used to determine the diversity gain (as the SIR goes to infinity), and it turns out that there is no diversity gain in simple retransmission schemes, even with independent Rayleigh fading over all links. We also determine the complete joint SIR distribution for two transmissions and the distribution of the local delay, which is the time until a repeated transmission over the reference link succeeds.

Wireless networks, interference, correlation, outage, Poisson point process, stochastic geometry.

## I Introduction

### I-a Motivation and contributions

The locations of interfering transmitters in a wireless network are static or subject to a finite level of mobility. As a result, the interference power is temporally correlated, even if the transmitters are chosen independently at random from the total set of nodes in each slot. The interference correlation has been largely ignored until recently, although it can have a drastic effect on the network performance. In this paper, we provide a comprehensive analysis of the joint success and outage probabilities of multiple transmissions over a reference link in a Poisson network, where the potential interferers form a static Poisson point process (PPP) and the actual (active) interferers in each time slot are chosen by an ALOHA multiple-access control (MAC) scheme. The results show that for some network parameters, ignoring interference correlation leads to significant errors in the throughput and delay performance of the link under consideration.

The Poisson network model has served as an important base-line model for ad hoc and sensor networks for several decades and later also for mesh and cognitive networks. More recently, it has also been gaining relevance for cellular systems, where base stations are increasingly irregularly deployed, in particular in heterogeneous networks [1]. Consequently, the results in this paper may find applications in a variety of networks.

The paper makes four contributions:

• We introduce the diversity polynomial and provide a closed-form expression for the joint success probability of transmissions in a Poisson field of interferers with independent Rayleigh fading and ALOHA channel access (Section III).

• We show that there is no temporal diversity gain (due to retransmission), irrespective of the number of retransmissions—in stark contrast to the case of independent interference (Section III.D).

• We provide the complete joint SIR distribution for the case of two transmissions and show that the probability of succeeding at least once is minimized if the two transmissions occur at the same rate (Section IV).

• We determine the complete distribution of the local delay, which is the time it takes for a node to transmit a packet to a neighboring node if a failed transmission is repeated until it succeeds (Section V).

### I-B Related work

The first paper explicitly addressing the interference correlation in wireless networks is [2], where the spatio-temporal correlation coefficient of the interference in a Poisson network is calculated. It is also shown that transmission success events and outage events are positively correlated, but their joint probability is not explicitly calculated. In [3], the temporal interference correlation coefficient is determined for more general network models, including the cases of static and random node locations that are known or unknown, channels without fading and fading with long coherence times, and different traffic models. In [4], the loss in diversity is established for a multi-antenna receiver in a Poisson field of interference. The probability that the SIR at antennas jointly exceeds a threshold is determined in closed form. This result is a special case of the main result in this paper, where the focus is on temporal correlation. More recently, [5] studied the benefits of cooperative relaying in correlated interference, for both selection combining and maximum ratio combining (MRC), while [6] analyzed on the impact correlated interference has on the performance of MRC at multi-antenna receivers.

A separate line of work focuses on the local delay, which is the time it takes for a node to connect to a nearby neighbor. The local delay, introduced in [7] and further investigated in [8, 9], is a sensitive indicator of correlations in the network. In [10] the two lines of work are combined and approximate joint temporal statistics of the interference are used to derive throughput and local delay results in the high-reliability regime. In [11], the mean local delay for ALOHA and frequency-hopping multiple access (FHMA) are compared, and it is shown that FHMA has comparable performance in the mean delay but is significantly more efficient than ALOHA in terms of the delay variance.

## Ii System Model

We consider a link in a Poisson field of interferers, where the (potential) interferers form a uniform Poisson point process (PPP) of intensity . The receiver under consideration is located at the origin , and it attempts to receive messages from a desired transmitter at location , where , which is not part of the PPP. Time is slotted, and the transmission over the link from to is subject to interference from the nodes in , which use ALOHA with transmit probability . The desired transmitter is transmitting in each time slot. The transmit power level at all nodes is fixed to , and the channels between all node pairs are subject to power-law path loss with exponent and independent (across time and space) Rayleigh fading.

The signal-to-interference ratio (SIR) at in time slot is then given by

 SIRk=hkr−α∑x∈Φkhx,k∥x∥−α,

where is the set of active interferers in time slot and is a family of independent and identically distributed (iid) exponential random variables with mean . In each time slot , forms a PPP of intensity , but the point processes and are dependent for all , since they are subsets of the same PPP . In the extreme case where , , . This dependence is what makes the following analysis non-trivial.

## Iii The Diversity Polynomial and the Joint Success Probability

### Iii-a Main result

We use a standard SIR threshold model for transmission success and denote by the transmission success event in time slot . We first focus on the probabilities of the joint success events

 p(n)s≜P(S1∩…∩Sn).

To calculate this probability, we introduce the diversity polynomial.

###### Definition 1 (Diversity polynomial).

The diversity polynomial is the multivariable polynomial (in and ) given by

 Dn(p,δ)≜n∑k=1(nk)(δ−1k−1)pk. (1)

It is of degree in and degree in .

The second binomial can be expressed as

 (δ−1k−1)≜(δ−1)⋅…⋅(δ−k+1)(k−1)!=Γ(δ)Γ(k)Γ(δ−k+1). (2)

The first four diversity polynomials are

 D1(p,δ) =p D2(p,δ) =2p+(δ−1)p2 D3(p,δ) =3p+3(δ−1)p2+12(δ−1)(δ−2)p3 D4(p,δ) =4p+6(δ−1)p2+2(δ−1)(δ−2)p3+ 16(δ−1)(δ−2)(δ−3)p4.

Properties:

• For fixed and , is concave increasing from to , for .

• For fixed and , is convex increasing from to , for .

###### Theorem 1 (Joint success probability).

The probability that in a Poisson field of interferers a transmission over distance succeeds times in a row is given by

 p(n)s=e−ΔDn(p,δ),

where and .

Proof. See Appendix A.
Remarks:

• The parameter is related to the spatial contention parameter introduced in [12, 13]. For Poisson networks, , hence .

• When evaluating as a function of , it must be considered that is itself a function of , not just .

• For , the result reduces to the well-known single-transmission result , for all .

• If (), , which means the success events become independent. At the same time, , so .

• If (), , which is the case of maximum correlation. At the same time, , which is the smallest possible value.

• If and , for all , so the success events are fully correlated (despite the iid Rayleigh fading), i.e.,

 p(1)s=p(2)s=…=e−Δ=e−λπr2,

and . This is a strict hard-core condition, i.e., all transmissions succeed if there is no interferer within distance .

• If , the diversity polynomial simplifies to the one introduced in [4] for the SIMO case, where it quantifies the spatial diversity instead of the temporal diversity:

 Dn(1,δ)=Γ(n+δ)Γ(n)Γ(1+δ)

As these remarks show, the diversity polynomial characterizes the dependence between the success events and the diversity achievable with multiple transmissions.

An immediate important consequence of Thm. 1 is the following result for the conditional success probability of succeeding at time after having succeeded times:

 P(Sn+1∣S1,…,Sn)=eΔ(Dn(p,δ)−Dn+1(p,δ)). (3)

Fig. 2 displays the conditional success probability for . It can be seen that succeeding once or twice drastically increases the success probability if is not too small. This illustrates that treating interference as independent may result in significant errors.

### Iii-B Alternative forms of the diversity polynomial

Let

 fn(x)≜n∏k=1(xk−1)=1n!n∏k=1(x−k)

be the polynomial of order with roots at and . Thus equipped, we can write the diversity polynomial as

 Dn(p,δ)=n∑k=1(nk)fk−1(δ)pk,

by observing that

 fk−1(δ)=Γ(δ)Γ(k)Γ(δ−k+1).

Using the Stirling numbers of the first kind , the falling factorial111 is the Pochhammer notation for the falling factorial. can be written as

 (x)n=n∑k=0Sn,kxk.

Rewriting the binomial as

 (δ−1k−1)=(δ−1)k−1Γ(k)=1Γ(k)k−1∑j=0Sk−1,j(δ−1)j,

we have

 Dn(p,δ) =n∑k=1(nk)pkΓ(k)k−1∑j=0Sk−1,j(δ−1)j =n∑k=1(nk)pkΓ(k)k−1∑j=0(−1)jSk−1,j(1−δ)j. (4)

This expansion in is useful since in most situations. For , the polynomial in this form is

 D2(p,δ) =2p−p2(1−δ) D3(p,δ) =3p+(−3p2+12p3)(1−δ)+12p3(1−δ)2 D4(p,δ) =4p+(−6p2+2p3−13p4)(1−δ)+ (2p3−12p4)(1−δ)2−16p4(1−δ)3.

For , since , , we have from (4)

 Dn(p,δ)=np+(1−δ)n∑k=2(nk)(−1)k+1pkk−1+O((1−δ)2).

This expression is useful as an approximation for general if (or ).

Alternatively, can be expressed as a polynomial in as

 Dn(p,δ)=n∑k=1(nk)pkΓ(k)k∑j=1Sk,jδj−1.

In this last expression, the term for is . This is the polynomial in obtained when . Conversely, when , it is .

### Iii-C Event correlation coefficients

Let be the indicator that occurs. The correlation coefficient between and , , is

 ζAi,Aj(p,δ) =eΔp2(1−δ)−1eΔp−1. (5)

The correlation coefficient for is illustrated in Fig. 3 as a function of and . It reaches its maximum of at and . While it is decreasing in , it is not monotonic in at .

Since , we have , thus the failure events are correlated in exactly the same way as the success events: If , then .

### Iii-D Joint and conditional outage probabilities

The dependence between two success events can be quantified by the ratio of the probabilities of the joint event to the probability of the same events if they were independent. We obtain

 P(S1∩S2)P2(S1)=e−ΔD2(p,δ)e−2Δp=eΔ(1−δ)p2>1,

which is consistent with the fact that the correlation coefficient (5) is positive. The positive correlation is also apparent from the conditional probability that the second attempt succeeds when the first one did, which is

 P(S2∣S1)=e−ΔD2(p,δ)e−Δp=e−Δp(1−p(1−δ)).

The probability of (at least) one successful transmission in attempts follows from the inclusion-exclusion formula

 p1∣ns≜P(n⋃k=1Sk)=n∑k=1(−1)k+1(nk)p(k)s. (6)

For the joint outage it follows that

 P(¯S1∩¯S2)=1−p1∣2s=1−2e−Δp+e−Δp(2−p(1−δ)).

Hence

 P(¯S1∣¯S2)=1−e−Δp(1−e−Δp(1−p(1−δ)))1−e−Δp (7)

and

 P(¯S1∩¯S2)P2(¯S1)=1+e−2Δp(eΔp2(1−δ)−1)(1−e−Δp)2>1,

which is consistent with the previous observation that failure events are also positively correlated.

From (7), the success probability given a failure follows as

 P(S2∣¯S1)=1−e−Δp(1−p(1−δ))eΔp−1,

which is maximized at , where it is .222Here and elsewhere in the paper, we assume that when a function has a removable singularity at , its value at is understood as the limit . This follows since the numerator is at most whereas the denominator is at least , both with equality at . This yields the general bound

 P(S2∣¯S1)≤1−p(1−δ),

with equality if and only if . Since , is achieved by either letting the interferer density , the transmission distance , or the SIR threshold go to .

Next we examine the conditional outage probability of an outage in slot given that outages occurred in slots through . Since as , one would expect this conditional outage probability to go to zero in the limit. Interestingly, this is not the case.

###### Corollary 1 (Asymptotic conditional outage).
 limΔ→0P(¯Sn+1∣¯S1∩…∩¯Sn)=p(1−δ/n),n≥1. (8)
###### Proof:

From (11) we know that the expansions of and both have non-zero linear terms in , thus the higher-order terms do not matter, and the limit follows as

 limΔ→0p(n+1)op(n)o=pnΓ(n+1−δ)Γ(n−δ)=pn(n−δ).

This is in stark contrast to the independent case, where this limit is obviously . The actual asymptotic conditional outage probability is increasing in and reaches as .

Conversely, we have for the conditional success probability given failures

 limΔ→0P(Sn+1∣¯S1∩…∩¯Sn)=1−p(1−δ/n).

Fig. 4 illustrates the conditional outage probability after failures for and .

### Iii-E Diversity gain of retransmission scheme

###### Definition 2 (Diversity gain of retransmission scheme).

The diversity gain, or simply diversity, is defined as

 d≜−lim¯¯¯¯¯¯¯¯SIR→∞logp(n)o(¯¯¯¯¯¯¯¯¯SIR)log¯¯¯¯¯¯¯¯¯SIR,

where is the mean SIR (averaged over the fading).

This is analogous to the standard definition in noise-limited systems, where diversity is defined as the exponent of the error probability as the (mean) SNR increases to infinity, see, e.g., [14]. In our interference-limited system, the relevant quantity is the SIR.

To calculate the diversity, we need to first establish the connection between the mean SIR and the parameter . The can be increased by either increasing the received signal power or by decreasing the interference. Either way, we find that :

• If we increase the received power by increasing the transmit power at the desired transmitter, we have . Since increasing and decreasing have the same effect on the success probability , we have and thus .

• If we increase the received power by reducing the link distance , we have . Since , we obtain .

• If we reduce the interference by decreasing the intensity of the PPP, we have since the interference is a stable random variable with characteristic exponent [15, Cor. 5.4]. Since and , we again have .

In conclusion, letting is the same as letting , and we can express the diversity as

 d=−limΔ−1/δ→∞logp(n)o(Δ)log(Δ−1/δ)=limΔ→0δlogp(n)o(Δ)logΔ. (9)

Next we need a lemma that establishes expansions on the probability of succeeding at least once in transmissions.

###### Lemma 1 (Taylor expansions).

We have

 p1∣ns=1−ΔpnΓ(n−δ)Γ(n)Γ(1−δ)+O(pn+1),p→0. (10)

and

 p1∣ns=1−ΔpnΓ(n−δ)Γ(n)Γ(1−δ)+O(Δ2),Δ→0. (11)

Proof: See Appendix B.

###### Corollary 2 (Diversity gain).

We have for all .

###### Proof:

From (11) we have for some that does not depend on . It follows that

 d=limΔ→0δlog(Δ(C+O(Δ)))logΔ=δ.

In contrast, with independent interference, the diversity gain would be

 limΔ→0δlog((1−e−Δp)n)logΔ=nδ.

So, retransmissions in (static) Poisson networks provide no diversity gain.

Conversely, fixing and varying , we have from (10) and the fact that

 limp→0δlogp(n)o(p)logp=δn,

so if the SIR is increased by decreasing , full diversity is restored. The difference in the behavior lies in the fact that captures the static components of the network, while reducing reduces the dependence between the interference power in different time slots.

Alternatively, the diversity could be defined on the basis of instead of , which would yield diversity in the independent case (and diversity in reality). This value may be better aligned with the intuition of what the diversity gain should be with independent transmission attempts.

### Iii-F Effect of bounded path gain

Here we derive the conditional success probability for the case where the (mean) path gain is bounded, i.e., instead of assuming a gain of for a link of distance , we employ a path gain of . Equivalently, the path loss is .

###### Corollary 3 (Joint success probability for bounded path gain).

For the same setting as in Thm. 1 but with path loss law , the joint success probability of transmissions over distance is

 p(n)s,bd=exp(−λπn∑k=1(−1)k+1pkBk), (12)

where

 Bk=(θ′1+θ′)k+θ′δδΓ(k−δ)Γ(δ)Γ(k)−H([k,δ],1+δ,−1/θ′),

is the Gauss hypergeometric function333Sometimes denoted as , and .
Compared with the unbounded case in Thm. 1, we have if .

Proof. See Appendix C.
The middle term in the expression for is the one for the unbounded path gain, whereas the other two account for the difference between the unbounded and bounded case. Since , the bounded and unbounded cases coincide as , i.e., for large SIR thresholds or distances of the desired link. Even for and , the difference is insignificant, as Fig. 5 illustrates. The figure replicates Fig. 2 for bounded path gain and shows the same behavior: Succeeding once or twice significantly increases the success probability for not too small. This suggests that the conclusions and trends observed in the unbounded case also hold in the bounded case.

## Iv The Two-Transmission Case with Different SIR Thresholds

Here we explore the case of but with different thresholds, i.e., we focus on the events and . This case is of interest for two reasons: First it leads directly to the complete joint SIR distribution, second it is useful to provide guidance on how the rate of transmission affects the probabilities of succeeding twice or succeeding after a failure.

### Iv-a Main result

###### Theorem 2 (Joint success probability with different thresholds).

We have

 P(S1∩S2)=e−^Δ^D(p,δ,θ1,θ2),

where and

 ^D(p,δ,θ1,θ2)=p(θδ1+θδ2)+p2θδ1θ2−θδ2θ1θ1−θ2. (13)

Alternatively, letting and , we have

 ^D(p,δ,¯θe−ν,¯θeν)=p¯θδ(2cosh(νδ)−psinh(ν(1−δ))sinhν). (14)

Moreover, achieves its minimum of at , i.e., the joint success probability is maximized at .

Proof. See Appendix D.

Since the joint success probability is symmetric in and , the expression (14) is even in , and it can be tightly bounded by its quadratic Taylor expansion

 ^D(p,δ,¯θe−ν,¯θeν)≳p¯θδ(2−p(1−δ)+δ[δ+p6(δ−1)(δ−2)]ν2). (15)

With independent interference, we would have . As expected,

 ^D(p,δ,¯θe−ν,¯θeν)

which shows that transmission success events are positively correlated for all thresholds , .

The joint SIR distribution follows from Thm. 2 as

 P2(θ1,θ2) ≜P(SIR1≤θ1,SIR2≤θ2) =1−e−^Δθδ1p−e−^Δθδ2p+e−^Δ^D(p,δ,θ1,θ2). (16)

Expressed differently,

 P2(¯θe−ν,¯θeν)=1−2exp(−^Δp¯θδcosh(νδ))cosh(^Δp¯θδsinh(νδ))+exp(−^Δp¯θδ[2cosh(νδ)−psinh(ν(1−δ))sinhν]) (17)

The next result shows that is an extremal point of the joint outage probability.

###### Corollary 4 (Asymmetric probability of success).

For all , , , , the probability of succeeding at least once in two transmissions with thresholds and , respectively, is minimized at , i.e., in the symmetric case.

Proof: See Appendix E.

Hence the probability of succeeding at least once in two transmissions can be increased by using asymmetric thresholds , corresponding to . Conversely, the joint outage probability is maximized at .

Since is an even function of , it can be expressed as

 p1∣2s(ν)=1−P2(¯θe−ν,¯θeν)=A+Bν2+O(ν4),

where and is the second derivative at . and are given by

 A =2P(SIR1>¯θ)−P(SIR1>¯θ,SIR2>¯θ) =2e−^Δp¯θδ−e−^Δp¯θδ(2−p(1−δ)) (18) B =^Δp¯θδδ2(^Δp¯θδ−1)e−^Δp¯θδ+ 16^Δp¯θδδ(6δ+2p−3pδ+pδ2)e−^Δp¯θδ(2−p(1−δ)). (19)

Since is the global minimum, we know that .

In Fig. 6, exact curves for and the quadratic approximations are shown for and . It can be observed that the approximation is quite accurate (slightly optimistic, in fact) for , which corresponds to .

### Iv-B Comparison with two independent transmissions

Here we investigate three cases where actual success probabilities are compared with the probabilities obtained if the two success events were independent.

#### Iv-B1 Joint success probability

Since transmission success events are positively correlated, we expect that the link can accommodate a certain level of asymmetry in the thresholds for the two transmissions. To explore this, we find the value of such that

 P(SIR1>¯θe−ν,SIR2>¯θeν)=P2(SIR1>¯θ)

or, writing out the probabilities,

 exp(−^Δ^D(p,δ,¯θe−ν,¯θeν))=e−2^Δ¯θδp.

To find an approximate value of for which this holds we use (15). Taking logarithms and dividing by yields

 2−p(1−δ)+δ[δ+p6(δ−1)(δ−2)]^ν2=2,

and we obtain

 ^ν2=p(1−δ)δ[δ+p6(δ−1)(δ−2)]. (20)

This is the level of SIR asymmetry that can be afforded thanks to the positive correlation. The resulting joint success probability will be slightly higher than , since (15) is a (tight) bound.

Assuming a transmission rate of nats/s/Hz for an SIR threshold of , which can be achieved if Gaussian signaling is employed, the positive correlation translates to a rate gain or throughput gain since

 log(1+¯θe−ν)+log(1+¯θeν)=log(1+2¯θcoshν+¯θ2)

is increasing in . Compared to the symmetric case, the throughput gain is

 log(1+2¯θ(coshν−1)(1+¯θ)2)≳log(1+¯θν2(1+¯θ)2).

#### Iv-B2 Probability of succeeding at least once

Alternatively, one may want to ensure that the probability of succeeding at least once in two transmissions is the same as in the independent case. This is guaranteed if

 1−P2(θ1,θ2)=1−(1−P(SIR1>θ1))2

or, equivalently,

 1−P2(¯θe−ν,¯θeν)=1−(1−P(SIR1>¯θe−ν))2. (21)

To solve this equation for , we approximate , which is valid since is the minimum of per Cor. 4 and the curvature given by in (19) is small444A numerical investigation shows that the second derivative achieves its maximum value of for and . For most parameters, is significantly smaller. For the ones in Fig. 6, for example, for and for .. Hence an approximate solution of (21) is given by

 e−νδ=−log(1−√1−p1∣2s(0))^Δp¯θδ. (22)

is calculated in (18) and denoted by .

In Fig. 7, the design procedure is illustrated. At , the probabilities and are shown in solid and dashed curves, respectively. First we observe that while independent transmission success would yield a success probability of at , the actual success probability is slightly less than . The two curves intersect at . So if a threshold of was used in the independent case and thresholds were used for the two transmissions in the dependent case, the success probability would be about 91% in both cases. So the penalty in the SIR threshold due to the correlation is about . This is the necessary reduction in the threshold for the second transmission to achieve the same two-transmission success probability as in the independent case.

Since the intersection between the solid and dashed curves cannot be calculated in closed form, the intersection between (the dash-dotted curve) is used instead, which yields the slightly conservative value of .

#### Iv-B3 Conditional success probability after failure

Lastly, one may want to choose the threshold for the second transmission such that the conditional success probability after a failure is still as large as the success probability in the independent case, i.e., the problem is to find such that

We have

 P(S2∣¯S1) =1−P(¯S2∣¯S1)=1−P2(θ1,θ2)P(¯S1) =e−^Δθδ2p−e−^Δ^D(p,δ,θ1,θ2)1−e−^Δθδ1p

This should be the same as . The resulting equation

 e−^Δpθδ2−e−^Δ^D(p,δ,θ1,θ2)=e−^Δpθδ1(1−e−^Δpθδ1)

can be numerically solved for .