Auto-Correlation and Coherence Time of Interference in Poisson Networks

The dynamics of interference over space and time influences the performance of wireless communication systems, yet its features are still not fully understood. This article analyzes the temporal dynamics of the interference in Poisson networks accounting for three key correlation sources: the location of nodes, the wireless channel, and the network traffic. We derive expressions for the auto-correlation function of interference. These are presented as a framework that enables us to arbitrarily combine the three correlation sources to match a wide range of interference scenarios. We then introduce the interference coherence time - analogously to the well-known channel coherence time - and analyze its features for each correlation source. We find that the coherence time behaves very different for the different interference scenarios considered and depends on the network parameters. Having accurate knowledge of the coherence time can thus be an important design input for protocols, e.g., retransmission and medium access control.


A. Motivation
The performance analysis of mobile communication and computing networks must model the interference caused by nodes on other nodes. Common models describe the average value or distribution of interference at a given receiver. More accurate modeling also considers the interference dynamics, i.e., the way interference changes over time and space [1]- [5]. Questions of interest are: How rapidly does interference change? What is the correlation of interference? Which parameters influence this correlation? Answers to these questions are particularly useful for the design of retransmission protocols and diversity schemes. Various researchers addressed these questions, taking different views.
A first view adopts measures like outage probability (or its counterpart, the coverage probability) for the purpose of protocol performance evaluation. This requires knowledge of the probability distribution of the signal-to-interference ratio (SIR), which in many cases is easier to calculate than some interference statistics [3]. It is, for example, possible to analyze the performance degradation or improvement of cooperative relaying [6]- [9] or MIMO [5], [10] impacted by interference.
A second view is to use interference dynamics to determine interference statistics, i.e., to establish the probability distribution of interference power in two or more points in time and/or space. Such multidimensional probability distribution is best represented by its probability density function. Unfortunately, for interference this function does not exist in closed form unless we consider scenarios with restricted network parameter values. Hence, we have to resort to characteristic functions, which are less flexible to handle but still valuable as they allow to calculate the moments of the distribution [3], [4] and other values.
Finally, a third view is to quantify the dependence of interference powers at different points in time and/or space. In other words, it addresses the question as to how much information is gained on the interference power (at a certain point in time and space) if we know interference power values close-by or from the past. Mathematically, this dependence is expressed in terms of correlation, with one application being interference prediction [11]: The auto-correlation function of the interference determines the order of the time series associated to the interference evolution over time, and can be the basis for designing a predictor.
The auto-correlation function of interference is still unknown. The reason is that most work on interference dynamics only considers the node locations as source of interference correlation. This assumption has two important consequences: First, the auto-correlation of interference at a given point in space is independent of the time lag τ , i.e., it is a constant function [12] not especially interesting to adopt in network performance analysis. Second, the resulting interference correlation may be inaccurate as important other sources of correlation are neglected. Two such additional sources of correlation are the data traffic and the wireless channel [13]. When considering these sources, the auto-correlation becomes non-constant and thus turns into an important tool for wireless network modeling and analysis.
In the article at hand, we generalize the known expressions for temporal correlation of interference in consecutive time slots (our previous work [13]) to arbitrary time lags τ . In other words, we calculate the auto-correlation function (ACF) of interference for this τ . Our work is based on the commonly used Poisson model, which means that the nodes of the network are distributed in space according to a Poisson point process. Random access to the wireless channel without channel sensing is used. As part of this work, we present modular expressions that allow different fading and traffic models to be incorporated by simply substituting a corresponding expression. In particular, we derive ACF expressions both individually for the three sources of correlation and for particular combinations of these sources. Furthermore, we analyze the coherence time of interference, which is the time until the interference correlation falls below a given threshold. We are able to derive closed form expressions in some cases and rely on numerical analysis in others.

B. Related work
There are several results on the temporal correlation of interference in Poisson networks (e.g., [5], [12]- [16]). All these results, however, consider only the node locations as source of correlation and assume a static network without mobility. Because of these assumptions the temporal correlation does not change over time, i.e., it does not depend on the time lag τ between the two instants under consideration. As a natural consequence, neither auto-correlation nor coherence time is considered.
Furthermore, results are available on the stochastic dependence of interference, which is typically expressed as the joint outage probability of several transmissions (e.g., [3], [4], [6]- [8], [17]). In all these publications, again the joint outage probability is independent of the time instant of the transmissions, which is a consequence of assuming that the node locations are the only source of correlation and that nodes are not mobile. Important exceptions to these limitations are the results by Gong and Haenggi [18], [19]. They also consider the node locations as the sole source of interference correlation but use a mobile network. It is the mobility that makes the temporal correlation to decrease for longer time lags τ . The analysis is performed for four different stochastic mobility models including Brownian motion (also adopted in the article at hand). The dependence of the correlation on the system parameters is analyzed with special focus on the average speed of the nodes. The evolution of the correlation in terms of the time lag is not extensively analyzed in those two publications and thus no notion of coherence time is considered.
The three sources of interference investigated in the article at hand are used in [13], where all 27 possible combinations of them are systematically addressed. However, the temporal correlation is only calculated for consecutive time slots, which provides no insights on the time it takes to reach low or negative correlation values.

C. Main contributions
We investigate the temporal dynamics of the interference in terms of its auto-correlation function and coherence time. We do so by accounting for three key sources of interference correlation: the node locations, the wireless channel, and the network traffic. Prior to this work, temporal correlation was only known for consecutive time slots [13], and for longer lags, only when considering the node locations as sole source of correlation. Only two possible outcomes apply: If there is no mobility among the nodes, the temporal correlation does not change over time [12], while for mobility it decreases [18] with increasing lags.
Our contributions can be summarized as follows: • We are the first to derive expressions for the autocorrelation function of interference in Poisson networks for the three key sources of correlation. These provide insights on the temporal dynamics of interference far beyond existing results, and are relevant for facilitating the exploitation of the temporal features of interference in emerging wireless systems. • The mathematical framework provided enables us to combine the three sources of correlation into a general expression for the temporal correlation of interference. Hence, unlike in previous work [13], it is no longer needed to address all possible combinations of the sources individually. Results are much more flexible and easier to apply in further work. • The analysis of the interference coherence time is further contributing to the practical applicability of our theoretical work. It provides an answer to the question as to how long a retransmission protocol has to wait for an uncorrelated channel. Our expressions also cover the fundamental and well-known channel coherence time without interference, which corresponds to taking only the channel into account as a source of correlation. Under this assumption, the interference coherence time and the channel coherence time are equal.
The rest of the article is organized as follows: Section II introduces the modeling assumptions. Section III provides general expressions for the temporal correlation of interference, which need specific sub-expression that model the different sources of interference correlation to be substituted. In Section IV the expressions that have to be substituted into the equations for temporal interference correlation are derived. Based on these results, the coherence time of interference is defined, and expressions for it are derived and used to analyze its features for different parameters in Section V. Finally, Section VI concludes the paper.

A. Node location and traffic
We consider a Poisson network: a wireless network consisting of nodes randomly located in a plane according to a Poisson point process (PPP) Φ on R 2 . The medium access opportunities are arranged into time slots. In each slot, every idle node decides independently from other nodes whether or not to start a new transmission. The duration of this transmission is d ∈ N time slots (message length), which is constant for all nodes and over time. We intend to have on average a fraction p of all nodes in Φ start a new transmission in each slot. Since only idle nodes start new transmissions, they adopt a sending probability of p 1−p(d−1) . Let q denote the probability that a node stays idle, i.e., q = 1 − and S t denote the set of all sending nodes at time t. The expected fraction of nodes sending in a given time slot t is thus µ = |S t | = p d, which we refer to as the traffic intensity.

B. Node mobility
Both static and mobile nodes are investigated. For mobile nodes, we letv denote the average speed of all nodes and consider two mobility models: linear mobility and timediscrete Brownian motion. The linear mobility model assumes the location of each node x at times t and t + τ to change in a random direction with |x t − x t+τ | =vτ , i.e., the distance increases linearly with time and we set ω τ = τ . This can be considered to be a reasonable model for time spans covering the duration of a few time slots, as the direction of movement and speed does not change significantly within the timescale of a slot in a practical system.
The location of a node x with Brownian motion at time t + 1 is [18], [19] x t+1 = x t +vω t , where ω t is a two-dimensional Gaussian random variable ω t ∼ N ( 0, Σ) with covariance matrix The random variables ω are i.i.d. for each time t. Hence, the location after τ slots is where d = denotes the equality in distribution. Remark: The homogeneity of the PPP describing the node locations is not altered by these two mobility models. At any time t the location of nodes forms a PPP with intensity λ. Note that this preservation of homogeneity is not provided by many other mobility models, for example, for the random waypoint model (nodes asymptotically concentrate in the center of the deployment area [20]).

C. Wireless channel
The wireless channel is modeled by a distance dependent path loss and multi-path fading accounting for reflections, diffraction, and other small-scale propagation effects. The signal power at a receiver x from an active sender y, with x, y ∈ R 2 is p RX = κ h 2 t xy .
In this equation, κ is the sending power of y, which we consider to be the same for all nodes in the network. h t models Nakagami-m block fading at time t and node x, i.e., h 2 t is gamma distributed according to h 2 t ∼ Γ(m, m). This implies E h 2 t = 1 and E h 4 t = m+1 m . The temporal aspect of fading is modeled in the following way [13]: We consider a block fading model, where the channel is assumed to remain constant over a duration of c ∈ N time slots after which it changes to an independent state, i.e., a random experiment is carried out independently of the previous channel state to establish the new channel state. This model for the temporal behavior is of widespread use. It matches well practical systems where the signal timing is usually designed to approximately meet this condition for easing the channel state acquisition and equalization tasks. Finally, xy = ( y − x ) denotes a nonsingular distance dependent path loss, for which we adopt with a path loss exponent α > 2.
be the path loss from a node x to the origin o.

D. Interference
Interference at time t is measured at the origin o of the plane R 2 , which is equal to the interference experienced by a typical node of the network due to Slivnyak's theorem. Its power is the sum of the signal powers of all sending nodes in the network (besides the intended signal from an specific sender, which is not considered in this work) yielding where γ t is a Bernoulli random variable indicating whether node x is sending (γ t = 1) at time t or not (γ t = 0).

E. Classification of correlation sources
We consider three sources of correlation of interference: node locations, wireless channel (i.e. correlated fading), and traffic. For each of them, there are three possible options, denoted by a triplet (i, j, k) ∈ {0, 1, 2} 3 : • They are constant or the correlation is not considered (denoted by 0). • They are random but uncorrelated (denoted by 1). • They are random and correlated (denoted by 2). This leads to 27 different cases that have been introduced in more detail and analyzed with respect to temporal correlation of interference in [13].

INTERFERENCE
We derive in the following paragraphs general expressions for the correlation of interference. These expressions are further specialized for selected interference scenarios in the next section. Interference correlation is measured in terms of Pearson's correlation coefficient denotes the covariance of interference at different time instants t 1 and t 2 , and var[I] = var[I t1 ] = var[I t2 ] denotes the variance of interference, which is constant over time due to the stationarity of the processes involved. The time lag from t 1 to t 2 is denoted by τ = t 2 − t 1 .
In the derivation of the general expressions, we distinguish between random and known node locations. In the former case we consider node locations as sources of interference correlation (cases (2, j, k)) while in the latter we do not consider them as a source (cases (0, j, k)).
A summary of the auto-correlation and the coherence time of interference without mobility is presented in Table I. The expressions in this table only hold for small time lags τ < c and τ < d. The more general expressions for arbitrary τ , and for mobility, are too long for the table but are available in the following sections. Coherence time results are computed by solving ρ(τ ) = 0 for τ . For the cases with '-' this can also be  I  SUMMARY OF RESULTS WITHOUT MOBILITY: AUTO-CORRELATION AND COHERENCE TIME OF INTERFERENCE. FOR THE COHERENCE TIME  EXPRESSIONS, ALL RESULTING NON-INTEGER VALUES HAVE TO BE ROUNDED TO THE NEXT HIGHER INTEGER. THE SYMBOL '-' DENOTES THAT WE   HAVE NO EXPRESSION FOR THIS CASE.   Locations Channel solved, but leads to expressions that have τ > c or τ > d for all parameters, which violates the presumptions for the simplified expressions of the auto-correlation functions presented in this table. Adopting the full expression for the auto-correlation does not lead to closed form expressions for coherence time.
Note that when substituting τ = 1 and m = 1 to the expressions in the table, the correlation results correspond to that in Table 1 of [13] with the following exception: In the cases (i, 2, 1) and (i, 2, 2) there is a difference due to a difference in the modeling assumptions. Here we assume that the channel of any pair of nodes is changing every c slots while in [13] the channel changes c slots after a transmission occurred.
A. Random node locations Theorem 1 (Correlation for cases (2, j, k)): The temporal correlation of interference between time instants t 1 and t 2 considering the node locations as a source of correlation is Proof: The expected value of interference is = dp λ where (a) holds due to Campbell's theorem [3], E h 2 t = 1 for all x ∈ Φ and t ∈ N and E[γ t ] = dp. The indices for the expectation operator E indicate the random variables involved and will be omitted in the future for shorter expressions. Aiming for the covariance, we calculate where we introduceh t andγ t to denote the fading coefficient and sending indicator of node y at time t, respectively. The first of the two expected values of (10) yields and the second gives In (a) we split the expected value for the different independent random variables and apply Campbell's theorem.
= µ 2 and the stationarity of the PPP. Hence, the covariance is The values of E h 2 t1 h 2 t2 and E[γ t1 γ t2 ] are characterizing the contribution of the wireless channel and the traffic to the correlation of interference, respectively. They depend on the values of j, k of the case (2, j, k) under consideration.
The variance is obtained by setting t 1 = t 2 in the above derivations yielding Dividing (13) by (14) yields the result. Corollary 1 (Correlation for cases (2, j, k) without mobility): The temporal correlation of interference between time instants t 1 and t 2 considering the node locations as sources of interference correlation and having no mobility Proof: Substitutingv = 0 into (8) yields the result.
B. Known node locations Theorem 2 (Correlation for cases (0, j, k)): The temporal correlation of interference between time instants t 1 and t 2 if neglecting the node locations as sources of interference correlation is Proof: The covariance of interference is The covariance in the second sum is always zero as the arguments are stochastically independent. The covariance in the first sum yields and in (b) we substituted E h 2 t = 1 and E[γ t ] = µ. Similar to the proof of Theorem 1, we calculate the variance by substituting t 2 = t 1 yielding Dividing (18) by (19) yields the result. Remark: • Since the node locations are not considered in Theorem 2, mobility is not taken into account in the result. • The correlation ρ[I t1 , I t2 ] = 0 for all cases (1, j, k).

INTERFERENCE
Using the general expressions derived in the previous section, we can now analyze the temporal correlation of interference for the three sources of correlation. We start by treating these correlation sources individually and afterwards look at some combinations that provide interesting insights. All plots have been compared to simulation results, which showed a good match. We refrain from plotting the simulation results as they would crowd the figures without providing any additional insights.

A. Correlation by node locations
The locations of the interfering nodes introduce a correlation that can be intuitively interpreted in the following way: If a receiver has close-by interferers, it is more likely to be disturbed in receiving a message than if no interferers are close-by. If there is no mobility at all, this correlation is independent of the time lag τ , i.e., ρ[I t1 , I t1+τ1 ] = ρ[I t1 , I t1+τ2 ] for all positive τ 1 , τ 2 ∈ N.
In case of mobility, the interference correlation decreases with time [19] depending on the average speedv and the type of mobility. Fig. 1 shows the temporal correlation over the time lag τ for both linear mobility and Brownian motion. For the same average speedv, the distance traveled after time τ is on average smaller in case of Brownian motion, and hence the correlation decreases slower with τ . In general, the decrease of correlation depends only on the distance traveled during the time lag τ or its distribution in case it is random (e.g. for Brownian motion).

B. Correlation by wireless channel
The wireless channel is modeled as a block fading channel with length c slots. This means that the channel gain due to multi-path propagation from a potential interferer stays unchanged for c slots and then changes to a stochastically independent value. This change is independent for each of the potential interferers. Hence, in each slot on average the channels of 1 d interferers change to a new state. This assumption introduces a correlation to the interference values of different slots for c > 1. In the expressions for interference correlation in Theorems 1 and 2, the effect of the channel of node x is covered by the term E h 2 t1 h 2 t2 . For Nakagami fading, this term depends on the fading parameter m, the channel block length c, and the time lag τ . It is given by In the special case of Rayleigh fading (m = 1), this term simplifies to E h 2 t1 h 2 t2 = 2 − τ c for τ < c. Remark: In case a node travels at least half the wavelength during a single time slot, the channel can be assumed stochastically independent for consecutive slots, i.e., c = 1. Under this assumption fading does not contribute to interference correlation and equations for cases (i, 1, k) for some i, j ∈ {0, 1, 2} apply.
When the wireless channel (i.e., fading) is the only source of interference correlation, we are in a static case (0, 2, 1). A plot of interference correlation in this case is shown in Fig. 2 over the time lag τ for different values of the channel block length c. The correlation decreases linearly with the time lag τ and vanishes for all τ ≥ c. For a given time lag, slower fading (higher values of c) implies a higher correlation. In the limit for a constant channel, i.e., c → ∞, we get independent of τ .

C. Correlation by data traffic
Correlated traffic is caused by having d > 1, which impacts interference correlation via the expected value E[γ t1 γ t2 ]. Lemma 1 (Probability of a node sending in two given time slots): The probability x sends in both slots t 1 and t 2 is . Proof: Let us assume that a certain node x is sending in both slots t 1 and t 2 . Then, there are two possibilities: (I) a single message could span both slots or (II) two different messages are transmitted in these two slots. A message consists of d time slots; we reference each of them by an index ranging over 1, 2, . . . , d. Let i denote the index of the message at time t 1 and j the index at time t 2 , and write the indices as tuples (i, j).
The probability P γ I t1,t2 that a single message spans over both t 1 and t 2 is given by This happens, as shown in Fig. 4, because in each time slot a fraction of p nodes start a transmission. For d > τ , there are d − τ slots for which a message starting there would span both time slots of interest. In this case, the indices (i, j) are always differing by the time lag j − i = τ . For d ≤ τ , the time difference between t 1 and t 2 is larger than the message length and hence it is impossible that a single message spans both slots. The probability P γ II t1,t2 of two different messages being transmitted at slots t 1 and t 2 is calculated by summing the probabilities of all possible indices (i, j) (see Fig. 3). The range of the index i is from max 1, d − (τ − 1) to d, i.e.,  1  2  3   1  2  3  1  2  3   1  2  3   1  2  3   1  2  3   1  2  3  1  2  3   1  2  3   1  2  3  1  2  3 Indices (1, 1)  if τ ≥ d, we have i = 1, . . . , d; in case τ < d, the index i has to be big enough to avoid a single message spanning both slots t 1 and t 2 . The range of the index j depends on the value of i, as the message covering slot t 2 must not overlap with the message covering t 1 . Hence, j ranges from 1 to min τ − (d − i), d , i.e., if there is enough space between t 1 and t 2 , j can go up to d; otherwise its maximum value is determined by the case where the two messages are transmitted directly one after another, as for the indices (1, 1), (2,2), and (3.3) in Fig. 3.
In order to calculate the probability of the situation described by an indices (i, j), we calculate the number of slots between the end of the message covering t 1 and the beginning of the message covering t 2 by These intermediate slots can be covered by additional messages in case there is enough space, i.e., if g ≥ d. The number k of messages fitting these intermediate slots is at most k ≤ g d slots, where x denotes the biggest integer that is smaller than or equal to x. If k messages are present in the intermediate slots, then there are e = g − kd slots unoccupied. The probability of a message starting is, as mentioned in Sec. II, 1 − q = p 1−p(d−1) , while the probability of an empty slot is q = 1 − p 1−p(d−1) . Therefore, the probability that t 1 and t 2 are occupied by different messages is Overall, we can sum the two probabilities calculated above to get the expected value E[γ t1 γ t2 ] = P γ I t1,t2 + P γ II t1,t2 . In (25) we substitute i by i − d to get the result.
Corollary 2 (Simplification for τ ≤ d): In the case the lag τ is smallter than or equalt to the message length, the result of Lemma 1 simplifies to where q = 1 − The inner sum of this expression is a geometric series (with the power being 0, . . . , τ − i − 1). After replacing the closed form result of the inner sum, the outer sum also results in a geometric series but with the first term (i = 0) missing. Applying the sum expression of geometric series twice yields where the 1 is due to the sum starting at i = 1 instead of 0.
Applying some basic algebra leads to the result. We investigate the temporal correlation of interference when the traffic is the only source of correlation (case (0, 0, 2)) with the aid of Fig. 5. It shows a heat map of the interference auto-correlation for different message lengths d. Correlation is highest for τ = 1 and decreases until the lag matches the message length (τ = d), where it is negative. For lags above d it increases to reach a small positive value, from where an oscillating behavior with reducing amplitude is observed. The lag for which zero crossings exist depend, besides the message length, on the sending probability p: for higher p the correlation is in general smaller which implies that it reaches zero for smaller τ and it gets more negative at τ = d.
A detailed study of the impact of p on correlation is presented in the two plots of Fig. 6. The first important observation is that the traces can be separated into two groups: The influence of p is different for d ≤ τ than for d > τ . For d ≤ τ the correlation is always negative if d mod 2 = τ mod 2 or otherwise mostly positive, only for small p it can take small negative values. Furthermore, it converges to zero for p → 0.
This behavior can be explained in the following way: Let us assume we have a message length d = 2. Since on average p nodes start a new transmission in each slot and they are chosen from the nodes that are idle, the nodes start to form two groups. One group of nodes start their transmissions in even slots while the other is starting transmissions in odd slots. This group formation is stronger for higher p. Hence, we have a negative correlation for even values of τ as mostly the same nodes transmit in t and t+τ , while we have positive correlation for odd values of τ as mostly nodes of different groups are transmitting in these two slots.
For d > τ there are significantly higher correlation values for small p and in the limit for p → 0 it approaches lim p→0 ρ[I t , I t+τ ] d>τ = d−τ d . For higher p the correlation decreases and becomes negative.
For all cases, we have lim p→ 1 d ρ[I t , I t+τ ] = 0, in which case all nodes are always transmitting and hence there is zero variance and covariance.
D. Correlation by multiple sources 1) Channel and traffic: Fig. 7 shows a heat map of the autocorrelation when both channel and traffic introduce correlation, which corresponds to case (0, 2, 2). Results are shown for c = 22. It can be noted that for given values of c and d the correlation is highest for τ = 1 and vanishes for high lags (at least τ > c, d). The correlation vanishes in the limit τ → ∞.  For each value of d there is a sharp change of trend at two points: The first is at τ = c and the second at τ = d. These are the points where the correlation caused by the traffic and by the channel, respectively, are at their minimum. The contribution of the channel is zero at τ = c and does not change for higher lags, while the contribution of traffic is negative for τ = d and increases when further increasing τ .
The case d = 10 is special since all nodes are transmitting all the time (i.e., we have a traffic intensity µ = 1). In such a setting the traffic does not cause any correlation and hence the correlation of interference is fully determined by the channel correlation. This corresponds to the case (0, 2, 0) for which the correlation shows a linear dependence on τ (topmost row in the heat map).
2) Node locations, channel, and traffic: When accounting for all three sources of interference correlation and no mobility (case (2, 2, 2)), the auto-correlation evolves as shown in Fig. 8. Correlation values start at a rather high level for small τ and decrease for higher τ , although not monotonically, as there are also ranges of τ where correlation slightly increases. For τ beyond the message length d and channel coherence time c, the correlation approaches its static value as determined by the node locations, i.e., case (2, 0, 1). In the limit τ → ∞ it converges to the values of that case. This is explained by noting that the contribution of the node locations to the correlation of interference does not change with τ . Hence, for values of τ for which the correlation contribution from other sources vanishes, the node locations are the only source of correlation and fully determine its value.
The specific trends of the auto-correlation in the plot are determined by the specific network parameters. The first qualitative change of the trend is the first local minimum, i.e. the point where it first starts to increase, which is located at τ = d. This corresponds to case (0, 0, 2), where a similar local minimum is present in the plot. The second qualitative change of the trend can be found at τ = c = 14, which is the time lag for which the wireless channel's correlation vanishes. Naturally, the order of these two events depends on which of the parameters d and c has a higher value. Overall, the correlation behaves similar to case (0, 2, 2), but for high τ in case (2, 2, 2) the correlation approaches a constant value while for case (0, 2, 2) the correlation approaches zero.

V. COHERENCE TIME
In the same style that the channel coherence time is defined in [21], we define the interference coherence time to be the time lag until the auto-correlation function of the interference becomes small and hence the interference becomes stochastically independent from its original value.
Definition 1 (Interference coherence time): The interference coherence time τ c is the minimum time lag τ such that the auto-correlation is smaller than a threshold θ, i.e., Remark: Note that this is a subjective definition since τ c is a function of θ, which has to be chosen in accordance to the considered case. The threshold below which the interference can be assumed to be uncorrelated is in general greater than zero. If, for example, a scenario with no mobility and correlation from the node positions is considered (case (2, j, k)), correlation will not drop to zero, no matter how high the time lag τ .
In general the coherence time depends on the sources of correlation and the time they need to uncorrelate. In the following we consider them separately to acquire an insight into their individual role.

A. Impact of traffic on coherence time
In this section we consider the coherence time τ c that results from a threshold θ = 0. If all transmissions span d slots, the correlation of interference caused by it monotonically decreases for d slots (see Fig. 5). However, the coherence time typically is shorter as after d slots the auto-correlation is negative and hence crossed zero earlier, i.e., τ c ≤ d. Hence, for the analysis of the coherence time we can adopt the simplified expression from Corollary 2.
In order to calculate the coherence time, we have to find the value of τ such that ρ[I t1 , I t2 ] = 0. For general parameters, there is usually no slot that exactly reaches zero correlation and therefore, we calculate instead the time lag τ until correlation reaches zero and then round it to the next higher integer.
Theorem 3: The coherence time when traffic is the only source of correlation (case (0, 0, 2)) is where q = 1 − p 1−p(d−1) and x = min{y ∈ N | y ≥ x} is the smallest integer being larger than or equal to x. Proof Solving the equation ρ[I t1 , I t2 ] = 0 for the time lag τ gives In general the solution of τ in this expression is a non-integer and hence we have to round it to the next higher integer as the correlation is monotonically decreasing with τ and we aim for a correlation being smaller or equal to zero. Remark: Although Theorem 3 is derived for no fading, the same expression holds for fading with c = 1, i.e., for case (0, 1, 2) as the only consequence of considering fading is to divide the auto-correlation function by a constant value m+1 m . Fig. 9 shows the corresponding plot of the interference coherence time. It shows that in case of very small sending probabilities p (close to zero) the coherence time is roughly equal to the message length (τ c ≈ d). For increasing values of p the coherence time is monotonically decreasing and approaches its minimum for µ → 1, which is lim p→ 1 d τ c = 1. Hence, in this case interference is already uncorrelated in consecutive slots.
It is interesting to notice that the coherence time depends on the sending probability p. In potential applications that require two uncorrelated slots, e.g. a retransmission protocol, the retransmission back-off interval (=time lag) has to be adjusted to the traffic load of the network. For higher traffic loads the back-off interval could be shortened based on this coherence time result, leading to a lower transmission delay at the nodes. There might be of course other reasons that prevent the back-off interval from being too short, but still this example illustrates the potential of having a better understanding of interference dynamics.

B. Impact of channel on coherence time
If the channel is the only source of correlation, i.e., case (0, 2, 1), assuming again θ = 0 we have the following result.
Theorem 4: The interference coherence time τ c if the channel is the only source of correlation (case (0, 2, 1)) equals the channel coherence time c.
Proof: From Theorem 2 we have that the correlation coefficient for the case (0, 2, 1) is where E h 2 t1 h 2 t2 is given in (20). For the case τ = c − 1 we have ρ[I t1 , I t2 ] = p c(1+m−mp)) , which is always positive for p > 0. As for τ ≥ c the correlation vanishes, the coherence time is always equal to c.

C. Impact of node locations on coherence time
If the node locations are considered as a source of correlation and a threshold θ = 0, it is important to assume mobility. Otherwise the temporal correlation caused by only node locations is constant for all τ ≥ 1 and never reaches θ. In case other sources of correlation exist, the correlation converges to this constant for τ → ∞ (actually the convergence is rather fast for reasonable values of c and d). Specifically, if the static node locations are the only source of correlation, we have ρ[I t1 , I t2 ] = p [12], [13], independent of τ . In such case it makes no sense to talk about a coherence time as for θ = 0 (or more generally for sufficiently small θ).
Let us assume that nodes move at an average speedv > 0. The temporal correlation of the interference is monotonically decreasing to its limit lim t→∞ ρ[I t1 , I t2 ] = 0. For finite time, however, it will get arbitrarily small but remain positive. Hence we choose a threshold θ > 0 for our analysis.
There exists no closed-form expression of the coherence time of interference τ c when correlation is induced by the node locations. This is due to the rightmost integration in (8) that to the best of our knowledge has no closed-form solution and therefore, cannot be rearranged to give an expression for τ . Accounting to this, we numerically evaluate the coherence time τ c . The numerical results are presented in Fig. 10, where the coherence time τ c for a threshold θ = 0.01 is investigated. The plot shows τ c for varying sending probability p and different average node speedsv. The general trend corresponds to intuition: Firstly, τ c increases with increasing p, as the temporal correlation of interference increases with p, making the threshold θ to be crossed later. For very small sending probabilities the coherence time is very small (in the limit we have lim p→0 τ c = 1), and the interference in consecutive slots is uncorrelated. Secondly, a higher average speedv of the nodes leads to a smaller coherence time. The reason for it is that with higher speed the nodes reach earlier the distance determining the decorrelation of interference.
The general trends are the same with Brownian motion, but the coherence times are higher, as for this mobility model the distance between the initial and final position of the nodes increases on average slower. This happens because of the back and forth movements of the nodes. We omitted a plot of these results as they do not provide further insights.

VI. CONCLUSIONS AND FUTURE WORK
We investigated the temporal dynamics of interference in Poisson networks with a focus on its auto-correlation function and coherence time. We see from the auto-correlation function that the correlation of interference for small time lags typically decreases. This decrease strongly depends on the sources of correlation. When evaluated over a large span of time lags, the correlation approaches zero, follows a damped oscillation around zero or shows a completely different behavior, all depending on the chosen scenario. It is evident that networking protocols and techniques have to be designed and parametrized differently for such different scenarios in order to operate best. Hence, a good knowledge of the auto-correlation function of interference contributes to improved network performance and robustness. For example, if interference traces are interpreted as time series, their underlying model order is determined by the auto-correlation function. Hence, our work contributes to model design by providing essential input parameters. It is thus a key enabling result for interference prediction based on time series.
The coherence time of interference can be calculated by means of expressions provided in the article at hand. This is a valuable tool for designing network protocols properly. Moreover, the expressions are simple enough to be implemented on networked nodes, allowing them to adapt to a changing network environment. This is a requirement in scenarios with, e.g., high mobility or high fluctuations in the network nodes.
Ongoing work analyzes how, by estimating a small number of network parameters, the node gathers a good estimate of the interference coherence time. Based on this information, we plan to devise a distributed algorithm to optimize the node's successful communication attempts and thus the overall network performance.