On Outage Probabilities With Correlated Locations, Channel, and Traffic in Wireless Networks

In this letter we derive novel expressions for the joint outage probability at two time slots in a wireless network under correlated interferer locations, channel gains, and traffic. The result is given as a function of the lag between the two slots, the channel coherence time, and the traffic burst length. Furthermore, we analyze the success probability of a transmission following a single outage, an important metric when, e.g., analyzing retransmission or cooperative relaying protocols.


I. INTRODUCTION
W HEN evaluating the performance of a wireless net- work, an important performance metric is the outage probability [1].It is often used when assessing the network's performance by mathematical expressions rather than by simulations or experiments.
Outage probabilities are usually derived using tools from stochastic geometry [1].A particular strength of this approach is that it can capture spatiotemporal correlations of interference [2], thus drawing a comprehensive picture of the performance of a network.In such an approach, the node locations are often modeled by Poisson point processes (PPPs), which is a good compromise between the accuracy of the model and mathematical simplicity and thus has given rise to a large number of publications, including results on assessing the performance of cellular networks [3], [4], mmWave communications [5], wireless ad hoc networks [6], and the Internet-of-Things (IoT) [7], [8].Many of these works take into account the interference correlation that is caused by the fixed interferers' locations or, in case of mobility, the correlation of the interferers' locations at different times.
However, as we have shown in the past, interference can have additional sources of correlation besides interferer locations, namely correlated channel gains (e.g., when fading has a coherence time that spans several slots) and correlated traffic [2], [9] (e.g., when it is especially likely that a given node transmits again after a transmission occurred due to bursty transmissions or the use of a retransmission protocol).These correlation sources lead to rather diverse interference Udo Schilcher is with the Institute of Networked and Embedded Systems, University of Klagenfurt, 9020 Klagenfurt, Austria (e-mail: udo.schilcher@aau.at).
Stavros Toumpis is with the Department of Informatics, Athens University of Economics and Business, 104 34 Athens, Greece (e-mail: toumpis@aueb.gr).
Digital Object Identifier 10.1109/LCOMM.2024.3381213dynamics.For example, when nodes are immobile, the correlation of interference levels at two different time slots that is caused by the fixed interferer locations is independent of the time lag between the two slots.In contrast, the correlation caused by correlated channel gains and traffic depends on the time lag [2] between the slots; this has different implications for protocol design and network parametrization.
Our particular contributions are as follows: • We derive closed-form expressions for the joint outage probability in two time slots separated by an arbitrary lag τ .These expressions, for the first time, quantify jointly the effects of correlations in the locations, the channel gains, and the transmission activities of the interferers.• We also derive closed-form expressions for the joint and the conditional (following a lost packet) success probabilities under the same conditions.• Having these closed-form expressions available, for the first time we quantify the manner in which the outage, joint success, and conditional success probabilities across two slots depend on the time separation of these slots, when there is no node mobility.Previous models did not capture this dependence.• We present several numerical results that quantify the impact of the aforementioned correlations on outage.• We provide guidelines for designing and configuring networking protocols, such as retransmission and scheduling schemes, cooperative relaying protocols, etc. • We illustrate similarities between interference correlation and joint outage probabilities, which validates using the correlation as an indicator of network performance.

II. SYSTEM MODEL
We consider a wireless network with nodes distributed according to a PPP Φ ⊆ R 2 with intensity λ.By applying Slivnyak's theorem, we assume, with no loss of generality, that the receiver under consideration r, the so-called typical receiver, is located at the origin o.An illustration of the setup is provided in Fig. 1.
Time is slotted and in each slot any idle node starts a new traffic burst of length d slots with probability p.Therefore, in each slot on average a fraction p of all nodes start a new transmission and overall a fraction pd ≤ 1 of the nodes are transmitting, while a fraction 1 − pd are idle, where p and p are connected by the equation p = All nodes transmit with unit power.The wireless channel is subject to distance-dependent path loss and Rayleigh fading.Hence, the reception power at time t from a node x ∈ Φ is ℓ x h x (t), where ℓ x = ∥x∥ −α with α > 2 is the path gain and the fading gain h x (t) is exponentially distributed with Fig. 1.Illustration of the network setup during a slot: The receiver r located at the center is aiming to receive data from a sender s.This reception is disturbed by several orange interferers x ∈ Φ.The gray nodes are idle during that slot.
mean E [h x (t)] = 1.We assume block fading, in which the channel state stays constant over c slots and then changes to independent value.All concurrent transmissions except the desired one cause interference.The interference power at time t is the sum of all powers of interfering transmissions arriving at r, i.e., x∈Φ ℓ x h x (t)γ x (t).Here, γ x (t) = 1 if x transmits at time t and γ x (t) = 0 otherwise.Thus, γ x (t) is a Bernoulli random variable with mean E [γ x (t)] = pd, which is the probability that x transmits at time t.The typical receiver r is aiming to receive transmissions from a sender s ̸ ∈ Φ at distance ∥s∥.For simpler notation, r, s, and any x ∈ Φ denote both the node and its location.The reception is successful if the signal-to-interference ratio SIR t at time t is above a threshold θ, i.e., otherwise r is in outage.We define the events and

A. Sending Probabilities
We start by deriving expressions for the probability that a given interferer is transmitting in the considered slots.
Proof: An interferer x transmits in both t 1 and t 2 in two different events: Either a single message spans both slots, or two separate messages overlap with the slots.Let P γ I x (t 1 , t 2 ) denote the probability of the first event.This probability vanishes for τ ≥ d.For τ < d, we have , since there are d − τ slots before t 1 in which a message can start that overlaps with t 2 , each with probability p. Next, let P γ II x (t 1 , t 2 ) denote the probability of the second event.The message overlapping with t 1 can start the earliest at d − 1 slots before t 1 and the latest at t 1 or d slots before t 2 , whichever comes first, as otherwise it would overlap with t 2 .Let i denote the number of slots this message extends after t 1 .Then, the message overlapping with t 2 can start the earliest at t 1 + i + 1, but not before t 2 − d + 1.Let j − 1 denote the number of slots that this message extends before t 2 .Then, we have i = 0, . . ., min(τ − 1, d − 1) and j = 1, . . ., min(τ − i, d).
The number of slots between the two messages in t 1 and t 2 is g = τ − i − j.If g < d, these slots are idle.Otherwise, there could be k ≤ ⌊ g d ⌋ messages in-between, where ⌊•⌋ is the floor operator.The number of idle slots is then e = g − dk.
The probability for given i, j, k Note that the power k + 1 is because of the probability that k messages start in-between, times the probability that the message of t 2 starts.Summing over these indices gives The binomial coefficient in the previous expression accounts for the number of sequences of messages and idle slots between the two messages in t 1 and t 2 .Adding P γ I x (t 1 , t 2 ) + P γ II x (t 1 , t 2 ) yields the result.Lemma 2: The probability that an interferer x transmits in exactly one of the slots t 1 and t 2 with where g = τ − i − 1.
Proof: The proof is similar to the one of Lemma 1, except that x must not send in t 2 .
The probability that an interferer x is not transmitting in any of the two slots is then

B. Success Probabilities
The well known success probability in a single slot is [1]: Due to block fading, for the sender and each of the interferers there are two options: either the channel states h x (t 1 ) and h x (t 2 ) are equal, or they are independent.These cases lead to different joint success probabilities.
1) Channel of Sender is Independent: We assume that the channel of the sender, i.e., h s (t 1 ), h s (t 2 ) are independent and identically distributed (i.i.d.).
Lemma 3 (Success for h x (t 1 ) = h x (t 2 )): The joint success probability if h x (t 1 ) = h x (t 2 ) for all x, but the coefficients h s (t 1 ), h s (t 2 ) of the sender are i.i.d., is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Proof: The joint success probability is where u = θ s ℓ x .In (a) we define and apply the complementary cumulative distribution function of the exponentially distributed h s (•).In (b) we apply the probability generating functional of Φ. Substituting p 00 = 1 − 2 p 10 + p 11 and solving the integral yields the result.Lemma 4 (Success for independent h x (t 1 ), h x (t 2 )): The joint outage probability if h x (t 1 ), h x (t 2 ) are i.i.d. for all x is Proof: Along the lines of the proof to Lemma 3, we have where u = θ s ℓ x .The difference to Lemma 3 is in the denominator of the last fraction.Solving yields the result.
Corollary 1: We assume a block fading channel with block length c for all interferers, and t 2 − t 1 = τ .Furthermore, let the channel states of the sender h s (t 1 ), h s (t 2 ) be i.i.d.In the case τ < c, we have For τ ≥ c the channels of all interferers become independent and the result of Lemma 4 applies.Proof: For block fading with length c, the expected fraction of interferers having an independent channel in the two slots is τ c for τ < c and 1 otherwise.The other interferers keep the same channel.Hence, the set of interferers having a dependent / independent channel form two independent PPPs [10].We combine them using a weighted sum of the expressions within the exponential function in Lemmata 3 and 4 yielding the result.
2) Channel of Sender is Constant: Lemma 5 (Success for h x (t 1 ) = h x (t 2 )): The joint success probability if h x (t 1 ) = h x (t 2 ) for all x and h s (t 1 ) = h s (t 2 ) is lower bounded by Proof: Along the lines of the proof to Lemma 3, we have where u = θ s ℓ x .Solving the integral yields the result.Lemma 6 (Success for independent h 1 , h 2 ): The joint success probability in time slots t 1 and t 2 if h x (t 1 ), h x (t 2 ) are independently exponentially distributed for all interferers and h s (t 1 ) = h s (t 2 ) is lower bounded by Proof: Along the lines of the proof to Lemma 5, we have where u = θ s ℓ x .Solving the integral yields the result.Corollary 2: We assume a block fading channel with block length c for all interferers and t 2 − t 1 = τ .Furthermore, let the channel of the sender be constant, i.e., h s (t 1 ) = h s (t 2 ).In the case τ < c, we have the lower bound For τ ≥ c the channels of all interferers become independent and the result of Lemma 6 applies.
Proof: The proof goes along the lines of the proof of Corollary 1, but applying Lemmata 5 and 6.

C. Conditional and Outage Probabilities
The outage probability for both slots t 1 and t 2 is The probability that the packet is successfully received at t 2 given that it is lost at t 1 is where P [S 1 ] is given in (5).The numerator is calculated by where P [S 1 S 2 ] is given in Corollary 1 in case of independent channels h s (t 1 ), h s (t 2 ) of the sender s.
In the case of h s (t 1 ) = h s (t 2 ), we can bound the conditional probability by applying Corollary 2, yielding IV. NUMERICAL RESULTS Fig. 2 shows P [O 1 O 2 ] using (16).The sending probability is normalized to pd, which is proportional to the expected interference power as on average a fraction pd of the nodes are transmitting.As expected, a higher pd implies a higher outage probability.In the domain d ≤ τ (the blue curves d = 1, 8, 10), we have a decrease of the outage probability with increasing d, while in the domain d > τ this trend is inverted, such that the lowest outage probability occurs for d = τ + 1.Moreover, the blue traces (d ≤ τ ) are below the black traces (d > τ ) for small pd, while for pd → 1 the blue traces increase and partly cross the black traces.At pd = 1, the trace d = 1 reaches a higher outage probability than all d > τ .This effect can be explained by a qualitative transition of interference correlation on p when d exceeds τ [2].Overall, d = τ shows the smallest outage probability over the whole plot.Furthermore, we compare the cases of independent and identically distributed (i.i.d.) and equal (h x (t 1 ) = h x (t 2 )) channel states of the sender.The outage probability is lower for i.i.d.channel states since when assuming there was an outage at t 1 , with an independent channel there is a higher chance that at t 2 there is success.Note that the traces of h x (t 1 ) = h x (t 2 ) are closer together than in the i.i.d.case due to the log-scale.
Fig. 3 investigates the relationship between the lag τ , the interference correlation ρ(τ ) (from [2], Case (2, 2, 2)), and the outage probability . This figure reveals several interesting insights: Firstly, the observation from Fig. 2 that outage is minimized for τ = d is confirmed.Secondly, the overall trend of the outage probability reflects the trend of the interference correlation.In particular, the correlation also has a minimum at τ = d.Furthermore, the bend of the correlation at τ = c is also seen in the outage traces.This emphasizes the importance of the knowledge about interference correlation,  such as results in [2], [9], [11], and [12].Thirdly, from a system design perspective, it is optimal when τ is chosen to be larger than the (expected) traffic burst length.In that way, the probability that both transmissions are in outage is improved.
In Fig. 4 we compare the joint success probability P [S 1 S 2 ] for constant and i.i.d.sender channel gains.The traces of the constant channel (blue curves) are lower bounds, while the i.i.d.traces (black curves) are exact.There is a significant difference between the two cases for a given τ , where the constant channel gives a higher success probability.The real gap might be even higher as we apply a lower bound for the constant channel.The difference reduces for higher τ , which seems natural.For analyzing a real network, it makes sense to apply the result of a constant channel for τ ≤ c, and the result of an i.i.d.channel for τ > c, based on the assumption that the channel of the sender and of the interferers c have the same coherence time.Considering these results from a network performance perspective, we can expect a jump of the success probability around τ = c.Hence, it is better to time transmissions of a given sender within a delay below c slots after a successful transmission to exploit the favorable conditions before they change to independent values.
In Fig. 5 we plot the conditional probability P [S 2 | O 1 ] versus p and τ .The conditional probability is always below the unconditional probability P [S 2 ], which indicates a positive correlation between the two slots.This correlation, and in turn the gap to the red line, varies with both p and τ .The correlation increases with p, which is a well known result [1], [9].However, the correlation has a more intricate relation with τ : for τ < d, the correlation decreases with increasing τ , leading to an increase in the conditional success probabilities Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.From a systems perspective we can conclude that after an outage it is best to send the next packet after τ = d.Furthermore, it can be observed that for rather high p the impact of τ decreases and vanishes close to pd = 1, where all nodes send all the time.This is also the point of the highest interference correlation.
In Fig. 6, we plot the conditional success probability P [S 2 | O 1 ] versus d and τ .The success probability is almost constant at a high value for d ≤ τ .For higher d the probability sharply drops to unfavorable values.This can be explained in terms of interference correlation: From Fig. 3 we see that for τ < d the correlation is much higher than for τ > d, where the traffic no longer contributes much.Hence, if d > τ , the high correlation makes a success following an outage less likely.For increasing τ , the dropping point is at higher d, which renders a higher τ better in terms of outage.However, it has to be considered that the higher τ increases the delay for message delivery and reduces throughput due to the longer gaps between consecutive transmissions, which forms a tradeoff.In practical applications, these gaps could still be used to transmit packets to different receivers, ideally located far away from the current destination.Again the success probability is maximized for the case d = τ .

V. CONCLUSION AND FUTURE WORK
In this letter, we derived closed-form expressions of the joint success, conditional success, and outage probabilities of two transmissions separated by a given time lag.We considered, for the first time, all three sources of interference correlation, i.e., interferer locations, correlated interferer channels, and correlated interferer traffic.We provided expressions for both equal and i.i.d.channels of the intended sender.Furthermore, we presented numerical studies of the expressions to highlight the difference with respect to the uncorrelated case as well as to the case when only interferer locations are a source of correlation.Finally, we drew conclusions that allow network designers to configure the network in a manner that takes into account the underlying conditions, such as how long is the optimal time lag between two consecutive transmissions.
Future work should explore three directions: Firstly, our results should be extended to an n-dimensional outage probability.Such a result is required, e.g., to analyze a scenario in which bursty traffic occurs, or where many retransmissions are required for a successful data delivery.Secondly, the results should be exploited to analyze and improve various protocols such as cooperative relaying.Thirdly, we could investigate how the senders can exploit channel state information (CSI) or an interference predictor to reduce outage by adjusting the timing / scheduling of their transmissions.

Manuscript received 31
January 2024; revised 4 March 2024; accepted 11 March 2024.Date of publication 25 March 2024; date of current version 12 June 2024.This work was supported by the Austrian Science Fund (FWF) under grant P24480-N15 and by the Hellenic Foundation for Research and Innovation under project HFRI-FM17-352.The associate editor coordinating the review of this letter and approving it for publication was M. Amadeo.(Corresponding author: Udo Schilcher.)

Fig. 2 .
Fig. 2. The joint outage probability of slots t 1 , t 2 (16) for different d is shown versus the sending probability pd.The upper group of lines is a lower bound for equal channel of the sender (hs(t 1 ) = hs(t 2 )) as in Corollary 2, while the lower group of lines is the exact probability for i.i.d.channel of the sender as in Corollary 1. Parameters are τ = 10, λ = 0.01, c = 10, α = 3, and θs = 1.

Fig. 5 .
Fig. 5.The conditional success probability of slot t 2 given an outage in slot t 1 of (17) for i.i.d.channel of the sender and different values of the lag τ versus the sending probability p. Black lines are for τ < d, while the blue lines are for τ ≥ d.The red line indicates the unconditional probability P [S 2 ] for reference.Parameters are λ = 0.01, c = 1, d = 9, α = 3, and θs = 1.