Predictive Ultra-Reliable Communication: A Survival Analysis Perspective
Sumudu Samarakoon, Mehdi Bennis, Walid Saad, Merouane Debbah
aa r X i v : . [ c s . N I] D ec Predictive Ultra-Reliable Communication: ASurvival Analysis Perspective
Sumudu Samarakoon,
Member IEEE , Mehdi Bennis,
Fellow IEEE ,Walid Saad,
Fellow IEEE and M´erouane Debbah,
Fellow IEEE
Abstract —Ultra-reliable communication (URC) is a key en-abler for supporting immersive and mission-critical 5G applica-tions. Meeting the strict reliability requirements of these appli-cations is challenging due to the absence of accurate statisticalmodels tailored to URC systems. In this letter, the wirelessconnectivity over dynamic channels is characterized via statisticallearning methods. In particular, model-based and data-driven learning approaches are proposed to estimate the non-blockingconnectivity statistics over a set of training samples with noknowledge on the dynamic channel statistics. Using principlesof survival analysis , the reliability of wireless connectivity ismeasured in terms of the probability of channel blocking events.Moreover, the maximum transmission duration for a givenreliable non-blocking connectivity is predicted in conjunctionwith the confidence of the inferred transmission duration. Resultsshow that the accuracy of detecting channel blocking eventsis higher using the model-based method for low to moderatereliability targets requiring low sample complexity. In contrast,the data-driven method shows higher detection accuracy forhigher reliability targets at the cost of × sample complexity. Index Terms —URC, channel blocking, survival analysis, sta-tistical learning, 5G.
I. I
NTRODUCTION
Next-generation wireless services, such as mission andsafety critical applications, require ultra-reliable communi-cation (URC) that provision certain level of communicationservices with guaranteed high reliability [1], [2]. Realizingthis in the absence of statistical models tailored to tail-centricURC systems is known to be a daunting task [3], [4].Towards enabling URC, the majority of the existing liter-ature relies on system-level simulations-based brute-force ap-proaches leveraging packet aggregation and spatial, frequency,and temporal diversity techniques [4], [5] while some assumeperfect or simplified/approximated models of the system (i.e.,stationary channel and traffic models) [6]. However, suchapproximations may fail to characterize the tail statistic ac-curately, and thus, may inadequate to fulfill the reliabilitytargets of URC [7]. In this view, machine learning (ML)techniques have been used in the context of URC includinglow-latency aspects with a focus on channel modeling andprediction [8]–[11]. These works are mostly data-driven and
This work is supported by Academy of Finland 6G Flagship (grantno. 318927) and project SMARTER, projects EU-ICT IntellIoT and EU-CHISTERA LearningEdge, Infotech-NOOR and NEGEIN, and the U.S.National Science Foundation under Grant CNS-1836802.Sumudu Samarakoon and Mehdi Bennis are with University of Oulu,Finland (e-mail: { sumudu.samarakoon,mehdi.bennis } @oulu.fi).Walid Saad is with ireless@VT, Bradley Department of Electrical and Com-puter Engineering, Virginia Tech, Blacksburg, VA (e-mail: [email protected]).M´erouane Debbah is with Universit´e Paris-Saclay, CNRS, CentraleSup´elec,91190, Gif-sur-Yvette, France (e-mail: [email protected])and the Lagrange Mathematical and Computing Research Center, 75007, Paris,France. assume the availability of large amounts of data. All priorworks focusing on channel modeling can be used to optimizetransmission parameters preventing communication outagesin terms of loss of received signal strength (RSS) due tochannel blockage. Here, a channel blocking event representsa period during which the RSS remains below a predefinedtarget threshold and the channel transitions from non-blockingto blocking events are analogous to the so-called survivaltime [12]. Characterizing such channel transitions is usefulto determine highly reliable transmission intervals under theabsence of knowledge of channel statistics, which has not beendone in the existing literature.The transitions between non-blocking and blocking canbe cast as lifetime events (birth-to-death) of the channels.Analyzing the time to an event (e.g., a channel transition)and rate of event occurrence are the prime focuses of survivalanalysis [13]. The applications of survival analysis span amultitude of disciplines including medicine (life expectancyand mortality rate from a disease), engineering (reliability ofa design/component), economics (dynamics of earnings andexpenses), and finance (financial distress analysis) [14]–[16].Therein, either model-based or model-free methods can beadopted. Hence, we adopt the analogy behind survival analysisto investigate non-blocking connectivity over wireless links.The main contribution of this work is to characterize thestatistics of non-blocking connectivity durations under the ab-sence of knowledge on the dynamic wireless channel statistics.In this view, we consider a simplified communication settingconsists of a single transmitter (Tx)-receiver (Rx) pair com-municating over dynamic channels with a fixed transmissionpower in order to characterize the transmission duration guar-anteeing a reliable non-blocking connectivity. The underlyingchallenge with the above analysis lies in assuming or acquiringthe full knowledge of non-blocking duration statistics, whichis unfeasible. Hence, we address two fundamental questions: i)how to accurately model the non-blocking duration statisticswithout the knowledge of channel statistics? and ii) how tocharacterize the confidence bounds for reliable transmissiondurations inferred from the devised non-blocking durationstatistics? To this end, we consider an exemplary scenario of abuyer named Buck who plans to purchase radio resources fora URC system from a seller named Seth. Here, Buck needsto evaluate the radio resources in terms of the transmissionperiods guaranteeing low blocking probabilities under differentconnectivity durations and the statistics of transmission periodsto enable URC. For this purpose, Seth wishes to reliablyevaluate the connectivity failure statistics, i.e., via survivalanalysis , using a set of non-blocking connected durationsamples M over dynamic channels. However, Seth mustaddress key questions related to the training data set M : i) observation period t blocking eventRSS threshold time (ms) R SS ( d B ) predictive period τ
550 650 750600 700 800-100-15-55 non-blockingevent
Fig. 1. An illustration of the channel blocking and non-blocking durationsfor a given RSS threshold. does it contain sufficient samples? ii) how confident am Iwith the reliability measures obtained using M ? and iii) is itbeneficial to improve the prediction confidence by investing inadditional sampling? Towards addressing these questions, wefirst cast the problem of finding the maximum transmissionduration yielding a predefined low blockage probability as anoptimization problem. Therein, we adopt a tractable parametricrepresentation for the probabilistic model of channel failures.To estimate the parameters, a minimization of a loss functionthat captures the gap between the true-yet-unknown channelfailure probability and the parametric representation is for-mulated. To minimize the aforementioned loss function, weadopt two approaches: a model-based approach that assumesa known prior probabilistic model following Weibull survivalanalysis, and a data-driven approach that uses function regres-sion via neural networks (NNs). For both techniques, wirelessconnectivity is analyzed in terms of the conditional failurestatistics, namely the statistics of the time to fail under givenconnectivity durations, and their confidence bounds followedby an evaluation based on simulations.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
Consider a one-way communication system in which a Txsends data to a Rx over a correlated flat fading channel.Due to channel and mobility dynamics, the RSS at the Rxfluctuates over time. For a given target RSS R , we definethe non-blocking connectivity probability (also called survivalprobability ) as Pr ( R t ≥ R ) where R t represents the RSS overthe duration [0 , t ] . In URC, the goal is to identify a predictiveperiod τ > that guarantees a low conditional blockingprobability after observing a non-blocking connectivity overa duration of t , i.e., Pr ( R τ + t < R | R t ≥ R ) ≤ ǫ given anoutage probability ǫ , as illustrated in Fig. 1.In this considered system, neither the channel dynamicsnor the statistics of non-blocking connectivity are known apriori. Our objective is to obtain a reliable measure of thecumulative density function (CDF) of the blocking events (orthe complementary cumulative density function (CCDF) ofthe connected durations), i.e., F ( t ) = Pr ( R t ≤ R ) . Once F ( t ) is characterized, the conditional failure probability at anobservation period t will be: Z t ( τ ) = Pr ( R τ + t < R | R t ≥ R ) = F ( t + τ ) − F ( t )1 − F ( t ) . (1)Then, determining the transmission duration followed by theobservation period of t for a given target reliability − ǫ , isformulated as follows: max τ, subject to Z t ( τ ) ≤ ǫ. (2) For a known and analytically tractable F ( · ) , the solution of(2) is given by τ ⋆ = F − (cid:0) ǫ + (1 − ǫ ) F ( t ) (cid:1) − t . However, F ( · ) is unknown due to the absence of channel statistics and thelack of accurate modeling of time-varying system parameters(e.g., network geometry, mobility, scattering coefficients, etc.),and thus, needs to be estimated.III. E STIMATING F ( · ) To estimate the non-blocking duration distribution, a para-metric representation of the CDF F θ ( · ) with parameter vector θ can be adopted. Here, θ is calculated using a set M of M connected duration samples. For this purpose, a loss function L ( · ) that captures the gap between the estimated and actualCDFs needs to be minimized over the sample set M asfollows: min θ L M ( F θ , F ) . (3)Towards solving (3), we consider two approaches: i) model-based approach: assuming a known prior probabilistic modelto derive the distribution parameters θ corresponding to theprior distribution using (3) and ii) data-driven approach: usingNN-based function regression over M where θ is the NNmodel to be learned from the data. A. Model-Based Approach
The events of non-blocking durations can be interpretedas the lifetimes of connected periods that are terminatedby the drop of RSS below a target threshold, which thenis followed by blocking events. In this view, the statisticaltools of survival analysis are suitable for characterizing thenon-blocking connectivity durations. In particular, Weibulldistribution is the most widely used lifetime data model dueto its relation to various families of distributions (uniform,exponential, Rayleigh, generalized extreme value, etc.) [13].Accordingly, the non-blocking connectivity durations can bemodeled by a Weibull distribution, F θ ( t ) = 1 − e − ( t/σ ) ξ , (4)where θ = ( σ, ξ ) is parameterized by the scale ( σ ) andshape ( ξ ) parameters. To find the most likely parameter valuesthat fit (4) to M , we use maximum likelihood estimation(MLE). In this regard, we define the loss function L M ( θ ) = − P m log f θ ( t m ) where f θ ( t ) = ξσ (cid:0) tσ (cid:1) ξ − e − ( t/σ ) ξ is theWeibull probability distribution function (PDF). Due to thenon-convex nature of the objective function, the estimatedparameters ˆ θ can be found using numerical methods (e.g.,stochastic gradient decent). Using ˆ θ , the failure probability in(1) becomes: Z t ( τ, ˆ θ ) = 1 − exp (cid:0) ( t/ ˆ σ ) ˆ ξ − (( t + τ ) / ˆ σ ) ˆ ξ (cid:1) . (5)Then, the solution for (2) will be: τ ⋆ = ˆ σ (cid:0) ( t/ ˆ σ ) ˆ ξ − ln(1 − ǫ ) (cid:1) / ˆ ξ − t. (6)Note that the reliable transmission duration τ ⋆ hinges onthe training data set M . Therefore, it is important to providethe margins of confidence for the derived values. To evaluate the confidence bounds, we adopt the likelihood ratio boundsmethod [17] given as: (cid:0) L M ( θ ) − L M (ˆ θ ) (cid:1) ≥ χ γ,M , (7)where χ γ,M are the Chi-squared statistics with probability γ and degree-of-freedom M , and θ is the unknown trueparameter, respectively. For example, γ = 0 . yields 95%confidence interval of the parameter estimation. Since weare interested in evaluating the confidence for τ ⋆ rather than θ = ( σ, ξ ) , we first find σ = ξ p ( t ξ − ( t + τ ) ξ ) / ln(1 − ǫ ) using (6) and, then, (7) can be modified as follows: L M ( ξ p ( t ξ − ( t + τ ) ξ ) / ln(1 − ǫ ) , ξ ) − L M (ˆ θ ) = χ γ,M . (8)Note that a closed-form expression cannot be derived for(8) which calls for numerical solutions (e.g., trust-regionalgorithm [18]). Since both τ and ξ are unknown in (8), forsome δ > , several priors for ξ from [ ˆ ξ − δ, ˆ ξ + δ ] are selectedfirst. By solving (8) for each of the above choices, a set ofsolutions { τ } is obtained, from which the confidence boundsof τ ⋆ are calculated. In addition to τ ⋆ , its mean and variancecan be analytically derived using (5). Proposition 1:
The N th moment of the non-blocking con-nectivity duration t + τ under the observation duration t is: E [( t + τ ) N ] = σ N e ( t/σ ) ξ Γ (cid:0) ( t/σ ) ξ ; 1 + N/ξ (cid:1) , (9)where Γ( α, β ) = R ∞ α x β − e − x dx is the upper incompletegamma function. Proof:
See Appendix A.Using the above result, the mean and variance of the remainingconnectivity durations at time t can be obtained from E [ t + τ ] − t and E [( t + τ ) ] − E [ t + τ ] , respectively. B. Data-Driven Approach
The main drawback of the model-based approach is itssusceptibility to model drift whereby the statistics of theactual observations may differ from the Weibull model. Hence,estimating F θ ( · ) by using the empirical distribution of samples M is preferable. Next, a data-driven approach based on a NN-based regression is presented.First, a subset of data samples M t = { t m | t m ≥ t, t m ∈M} is collected for a given observation period t . Then, theempirical distribution of the non-blocking duration samplesin M t is numerically evaluated so that a set of labeledtraining data tuples { ( t m , s m ) } are generated. Here, s m is theCDF value of t m calculated using the empirical distribution,which yields the corresponding failure distribution. The lossfunction is the mean square error (MSE) between the trueand estimated failure probabilistic values, i.e., L M t ( θ ) = M t P m (cid:0) s m − Z t ( t m , θ ) (cid:1) where Z t ( · , θ ) is modeled usinga multilayer perceptron (MLP) with model parameters θ . Tosolve (3), MLP uses ( t m , t m , . . . , t nm ) up to an order of n (toavoid under-fitting) as the input, s m as the output, and theMSE loss L M t ( θ ) as the empirical loss function. By trainingthe MLP in a supervised manner, Z t ( · , θ ) is derived. Finally, τ ⋆ that satisfies Z t = ǫ is obtained. Note that the accuracy of Z t ( · , θ ) relies on i) both quality and quantity of M t , ii) themodel complexity of θ , and iii) choice of the input size n . (a) Model-based estimation.(b) Data-driven estimation for different order of input sizes n ∈ { , } .Fig. 2. Comparison of the conditional failure probability estimation at t =0 . s for different sample complexities M ∈ { , , } . The N th raw moment of the remaining non-blocking con-nectivity for an observation duration t will be: E [ τ N | t ] = R ∞ τ N − (cid:0) − Z t ( τ, θ ) (cid:1) dτ. (10)First, the conditional probabilities are calculated from thetrained NN model over a sequence of τ = δk remainingconnectivity durations with k ∈ N and small δ > . Then, ap-proximating the integrations in (10) to numerical summations,the first and second moments of the remaining connectivitydurations can be obtained.IV. S IMULATION R ESULTS AND A NALYSIS
Here, we evaluate the characterization of non-blockingstatistics obtained via the proposed model-based and data-driven methods. For our simulations, we consider a timecorrelated Rayleigh flat fading channel model defined in [19].While we define a slotted time-based transmission with a slotduration of τ = 1 ms, for improved measurement accuracy,we consider a sampling frequency of kHz. For training,up to , non-blocking connectivity duration samples arecollected and for testing, additional , samples are used.Here, an RSS threshold of R = − dB is used for a unittransmit power. For the data-driven approach, we use an MLPwith two fully connected hidden layers with sizes of ten andsix and rectified linear unit (ReLU) activations. The outputlayer of the MLP is a single node with a symmetric saturatedlinear transfer function.Fig. 2 compare the conditional failure probability regressionperformance of both the model-based and the data-driven Fig. 3. Detection of blocked events based on the predicted duration τ ⋆ at t = 0 . s. approaches over the simulated data referred to as “simula-tion scenario” for different sample complexities, i.e., variouschoices of training sample sizes M ∈ { , ,
10 000 } .From Fig. 2a, we observe that the model-based design isalmost invariant over the choices of sample complexities dueto the accurate fit over probabilities above − . As theprobability decreases, the simulation results will deviate fromthe trend of higher probabilities. However, the model-basedmethod, which relies on the prior Weibull model, fails tocapture this deviation. In contrast, the data-driven regressionis susceptible to the lack of training samples as illustrated inFig. 2b. Moreover, it can learn the trends using data samplesand thus, the data-driven approach learns the low-probabilitybehavior of the simulation scenario as well. In addition, Fig.2b shows that increasing the order n from one to ten slightlyimproves the regression. This improvement is due to the factthat we consider the input as a tenth order polynomial of theconnectivity duration instead of order one.Fig. 3 compares the detection of channel blockingevents based on the predicted duration τ ⋆ from model-based and data-driven methods in terms of F-score : F = P TP P TP +( P FP + P FN ) / based on the events of true positive (TP) , false positive (FP) , and false negative (FN) [20]. We firstempirically partition the test connectivity durations dataset M ′ into two groups for a given reliability target (1 − ǫ ) : the positive group M + ǫ consisting of the smallest ǫ fraction of non-blocking durations and the rest composes the negative group M − ǫ . With this partitioning, for any test sample m ∈ M ′ there are three observation categories: i) TP : if m < τ ⋆ and m ∈ M + ǫ , ii) FP : if m < τ ⋆ but m ∈ M − ǫ , and iii) FN :if m ≥ τ ⋆ with m ∈ M + ǫ . In addition, for the purposeof comparison, a Gaussian process regression (GPR)-basedchannel estimation method proposed in [21] is adopted topredict consecutive non-blocking durations, which is referredto as the “GPR” baseline. Fig. 3 shows that as the samplecomplexity increases, the uncertainty of the estimated τ ⋆ de-creases and blocked events are accurately detected, achievinghigher F . For large ǫ , the estimated τ ⋆ from the model-basedapproach can accurately detect the channel blocking events(i.e., the lower tail) yielding high F . As ǫ decreases, themodel-based method based on the Weibull distribution biasdeviates from the actual data distribution even if the increasingtraining sample size M increases. From this result, we observethat the accuracy of channel blocking detection degrades by Fig. 4. Confidence limits of predictive transmission durations obtained usingthe model-based approach for observed durations t ∈ { , } ms. factors of × to × as shown in Fig. 3. In contrast, thedata-driven approach characterizes the lower tail better thanthe model-based method when a sufficiently large numberof training data is available. For a small M , the detectionaccuracy of the data-driven method approaches to zero withdecreasing ǫ , because of the lack of training data in the positiveset M + ǫ of the size of ǫM . Hence, increasing M = 100 to and then to
10 000 improves the blocked event detectionaccuracy from F = 0 to . and . at ǫ = 10 − and from F = 0 to . at ǫ = 10 − , respectively, highlighting theimportance of the sample complexity in data-driven methods.The GPR baseline outperforms both proposed methods with M ∈ { , } only for small reliability targets ǫ ≥ . .Due to the uncertainty in GPR, higher prediction errors can beobserved for tighter reliability targets, resulting in a low F .Fig. 4 illustrates the impact of sample complexity on theconfidence bounds of the predicted transmission durations at t ∈ { . , } s derived using the model-based approach. Here,a 95% confidence interval (i.e., γ = 0 . ) is used. FromFig. 4, we can see that MLE with few samples yields largeuncertainty in τ ⋆ while the uncertainty decays as M increasesdue to the monotonic decreasing nature of χ γ,M with M .This underscores the tradeoff between the model parameteruncertainty and the cost of data collection.The impact of the transmit power is investigated in Fig.5. Since R = − dB is used with a unit transmit power, a × and × increase in transmit power are captured with R of − dB and − dB, respectively. The effects of increasingtransmit power on the predicted connectivity durations derivedfrom the model-based approach are presented in Fig. 5a.Clearly, the non-blocking connectivity can be significantlyenhanced via increased transmission power.For a given observation duration t , the mean and varianceof the remaining non-blocking connectivity durations over thesimulated data and the estimations based on both the model-based and the data-driven methods are shown in Figs. 5b and5c, respectively. Note that the simulation scenario exhibitsdifferent trends at low and high t values and the number oftraining data samples reduces with increasing both t and R .Since the model-based approach is highly biased to the Weibullmodel, the accuracy of its mean and variance estimations ishigh only in the regimes where the majority of the trainingdata lies, and degrades with increasing t and R as illustrated (a) Impact of transmit power on the predicted trans-mission duration. (b) Expected time to fail for the observed data andpredictions. (c) Variance of failure time for the observed dataand predictions.Fig. 5. Impact of transmit power on the predicted duration ( τ ⋆ ) ensuring (1 − ǫ ) reliability (left), expected time to fail (middle), and its variance (right). in Figs. 5b and 5c. In contrast, due to having lower bias, thedata-driven approach generalizes throughout all t and R , butwith a price of significant accuracy losses in the mean andvariance estimations. V. C ONCLUSIONS
In this letter, we have analyzed the non-blocking connec-tivity of URC systems through the lens of model-based anddata-driven methods in order to estimate connectivity statisticsusing a set of non-blocking connectivity duration trainingsamples. Therein, we have measured the reliability of theconnectivity by using statistical tools from survival analysis.We have also validated our analysis based on simulations.The results show that the Weibull model-based method canbe accurately estimated with low sample complexity andcharacterizes well the tail events without the knowledge onthe channel statistics. In contrast, the data-driven design alignswell with the highly probable events under large sizes oftraining data highlighting the bias-variance tradeoff betweenthe aforementioned two approaches. Finally, this work pro-vides insights about the choice of transmit power in termsof channel blocking statistics. Future work will investigatehybrid approaches combining both data-driven and model-driven techniques. A
PPENDIX AP ROOF OF P ROPOSITION T = t + τ . By differentiating (5), the conditionalPDF is found as f t ( T ) = ξσ ξ T ξ − e − ( T/σ ) ξ e ( t/σ ) ξ for all T ≥ t . Then, the N th moment is given by E [ T N ] = R ∞ t ξσ ξ T N + ξ − e − ( T/σ ) ξ e ( t/σ ) ξ dT . Using the change of vari-ables with z = ( T /σ ) ξ and dT = σz /ξ − dz , E [ T N ] = R ∞ ( t/σ ) ξ σ N z N/ξ e − z e ( t/σ ) ξ dz, = σ N e ( t/σ ) ξ Γ (cid:0) ( t/σ ) ξ ; 1 + N/ξ (cid:1) , where Γ( α, β ) = R ∞ α x β − e − x dx is the upper incompletegamma function. R EFERENCES[1] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems:Applications, trends, technologies, and open research problems,”
IEEENetw. , vol. 34, no. 3, pp. 134–142, 2019. [2] J. Park, S. Samarakoon, H. Shiri, M. K. Abdel-Aziz, T. Nishio, A. El-gabli, and M. Bennis, “Extreme URLLC: Vision, challenges, and keyenablers,” arXiv preprint arXiv:2001.09683 , 2020.[3] P. Popovski, “Ultra-reliable communication in 5G wireless systems,” in
Proc. of 5GU , Akaslompolo, Finland, Nov. 2014, pp. 146–151.[4] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and low-latencywireless communication: Tail, risk, and scale,”
Proc. IEEE , vol. 106,no. 10, pp. 1834–1853, 2018.[5] C. Chaccour, M. N. Soorki, W. Saad, M. Bennis, and P. Popovski,“Can terahertz provide high-rate reliable low latency communicationsfor wireless VR?” arXiv preprint arXiv:2005.00536 , 2020.[6] S. Hur et al. , “Proposal on millimeter-wave channel modeling for 5Gcellular system,”
IEEE J. Sel. Topics in Signal Processing. , vol. 10, no. 3,pp. 454–469, 2016.[7] M. Angjelichinoski, K. F. Trillingsgaard, and P. Popovski, “A statisticallearning approach to ultra-reliable low latency communication,”
IEEETrans. Commun. , vol. 67, no. 7, pp. 5153–5166, 2019.[8] T. Nishio et al. , “Proactive received power prediction using machinelearning and depth images for mmWave networks,”
IEEE J. Sel. AreasCommun. , vol. 37, no. 11, pp. 2413–2427, 2019.[9] A. T. Z. Kasgari, W. Saad, M. Mozaffari, and H. V. Poor, “Experi-enced deep reinforcement learning with generative adversarial networks(GANs) for model-free ultra reliable low latency communication,” arXivpreprint arXiv:1911.03264 , 2019.[10] C. She et al. , “Deep learning for ultra-reliable and low-latency commu-nications in 6G networks,”
IEEE Network , vol. 34, no. 5, pp. 219–225,2020.[11] Z. Hou et al. , “Burstiness-aware bandwidth reservation for ultra-reliableand low-latency communications in tactile internet,”
IEEE J. Sel. AreasCommun. , vol. 36, no. 11, pp. 2401–2410, 2018.[12] 3GPP, “5G; service requirements for next generation new services andmarkets,” ETSI, Tech. Rep. 38.824 Rel-16, Mar. 2018.[13] R. B. Abernethy, J. E. Breneman, C. H. Medlin, and G. L. Reinman,“Weibull analysis handbook,” Pratt and Whitney West Palm beach flGovernment Products DIV, Tech. Rep., 1983.[14] J. Li and S. Ma,
Survival analysis in medicine and genetics . CRCPress, 2013.[15] S. M. Matz, L. G. Votta, and M. Malkawi, “Analysis of failure andrecovery rates in a wireless telecommunications system,” in
Proc. ofDSN , Bethesda, MD, USA, Jun. 2002, pp. 687–693.[16] E. K. Laitinen, “Survival analysis and financial distress prediction:Finnish evidence,”
Review of Accounting and Finance , 2005.[17] S. P. Verrill, R. A. Johnson, and Forest Products Laboratory (U.S.),
Confidence Bounds and Hypothesis Tests for Normal Distribution Co-efficients of Variation , ser. Research paper FPL-RP. USDA, ForestService, Forest Products Laboratory, 2007.[18] J. Nocedal and S. J. Wright,
Trust-Region Methods . New York, NY:Springer New York, 2006, pp. 66–100.[19] T. S. Rappaport,
Wireless communications: principles and practice .Prentice Hall, 1996, vol. 2.[20] D. C. Blair, “Information retrieval,”
J. Am. Soc. Inf. Sci. , vol. 30, no. 6,pp. 374–375, 1979, 2nd edition, London: Butterworths.[21] M. Karaca, T. Alpcan, and O. Ercetin, “Smart scheduling and feedbackallocation over non-stationary wireless channels,” in