Stochastic Dependence in Wireless Channel Capacity

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Markov Dependence delay, backlog, and delay-constrained capacity; second, the Let (Ω, F , (F t)t∈N , P ) be a filtered probability space and dependence is classified into three types, i.e., positive de- (Xt)t∈N be an adapted stochastic process. X is a Markov pendence, independence, and negative dependence, according process if and only if to the performance measure, it’s proved that the negative dependence is good to the channel performance, while the P (Xt ≤ x|Xt−1, Xt−2,..., X0)= P (Xt ≤ x|Xt−1) . (7) positive dependence does the opposite thing; last, the cause The Markov property is solely a dependence property that can of dependence in capacity is distinguished as uncontrollable be modeled exclusively in terms of copulas [5], [7]. (Copula parameters and controllable parameters, it’s elaborated how to is introduced in Appendix A.) use the controllable parameters to induce negative dependence The n-dimensional process X is a Markov process, if and to the capacity, and an example of using power as a control- only if, for all t < t <...

t t for m-dimensional copula A and n-dimensional copula B. S(s,t)= C(i) or S(s,t)= C(τ)dτ . (4) Z Proposition 1. If the dependence in capacity is driven by a i=Xs+1 s Markov process and the instantaneous capacity has a specific Denote S(t) ≡ S(0,t). The time average of the cumulative distribution with respect to a specific state transition, the capacity through [0,t) is defined as transient capacity: additive capacity together with the underlying Markov process, S(t) i.e., cumulative capacity, form a Markov additive process, C(t)= . (5) t which is a bivariate process with strong Markov property and the increments are conditionally independent given a Example 1. For a single input single output channel, if the realization of the underlying Markov process. channel side information is only known at the receiver, the instantaneous capacity is expressed as [3] In other words, a Markov additive process is a non- stationary additive process defined on a Markov process [13], C(t)= W log 1+ γ|h(t)|2 , (6) 2 [14], if the Markov process has only one state, it reduces where |h(t)| denotes the envelope of h(t), γ = P/N0W to a Markov process with additive increments. Since there denotes the average received SNR per complex degree of free- is no requirement on the 1-dimensional marginal distribution i dom, P denotes the average transmission power per complex Xt for X to be Markov, starting with a Markov process, a symbol, N0/2 denotes the power spectral density of AWGN, multitude of other Markov processes can be constructed by and W denotes the channel bandwidth. just modifying the marginal distributions [5], [7]. C. Distribution Bound A Markov additive process is defined as a bivariate Markov process {Xt} = {(Jt,S(t))} where {Jt} is a Markov process with state space E and the increments of {S(t)} are governed by {Jt} in the sense that [15] F E [f(S(t + s) − S(t))g(Jt+s)| t]= EJt,0 [f(S(s))g(Js)] . (12) We focus on the finite state space scenario and the structure is fully understood in discrete-time and continuous time settings. In discrete time, a Markov additive process is specified by the measure-valued matrix (kernel) F(dx) whose ijth element is the defective probability distribution

Fij (dx)= Pi,0(J1 = j, Y1 ∈ dx), (13) where Yt = S(t) − S(t − 1). An alternative description is in terms of the transition matrix P = (pij )i,j∈E , pij = Pi(J1 = j), and the probability measures Fig. 1. Transient capacity of Markov additive Rayleigh channel. According to the strong law of large numbers extended to the Markov additive process, Fij (dx) the transient capacity converges to the mean as time goes to infinity, i.e., the Hij (dx)= P (Y1 ∈ dx|J0 = i,J1 = j)= . (14) convergence of sample paths. The large deviation results are upper bound and pij lower bound with respect to the mean. Results are normalized, with violation probability ǫ = 10−3, W = 20kHz, SNR = [e0.5 e0.5; 0.7e0.5 0.7e0.5] With respect to a transition probability pij , the increment of and P = [0.3 0.7; 0.1 0.9], and 1000 sample paths. {St} has a distribution Bij . In continuous time, {Jt} is a Markov jump process specified With an exponential change of measure, a likelihood ratio by the intensity matrix Λ = (λij )i,j∈E . When {Jt} jumps process is formulated [16], from i to j 6= i, the jump of {St} has a probability qij (θ) with a distribution Bij . When Jt ≡ i, {St} evolves like a h (J ) t θS(t)−tκ(θ) (18) 2 L(t)= (θ) e , Lévy process with a characteristic triplet (νi,µi, σi ), where h (J0) νi ∈ R, σi ≥ 0, and νi is a nonnegative measure on R, ǫ which is a mean-one martingale. This martingale process if the Lévy measure ν satisfies, |x|ν (dx) < ∞ and i −ǫ i is useful for performance analysis of the wireless channel ν ([−ǫ,ǫ]c) = ν (dx) < ∞, ∀ǫ > 0, the Lévy i {x:|x|>ǫ} i R capacity. The distribution function results are as follows. exponent is expressedR as ∞ Theorem 1. For a Markov additive process, conditional on (i) 2 2 θx initial state , the distribution of the cumulative capacity is κ (θ)= θµi + θ σ /2+ e − 1 νi(dx), (15) J0 i Z ∞ expressed as, for some θ> 0, R(θ)S (θ) tκ(θ)−θx (−θ) tκ(−θ)+θx where θ ∈ Θ= {θ ∈ C : Ee 1 < ∞}. h (J0)e h (J0)e θS(t) 1 − (θ) ≤ FS(t)(x) ≤ (−θ) , Consider the matrix Ft[θ] = (Ei[e ; Jt = j])i,j∈E . In min(h (Jj )) min(h (Jj )) j∈E j∈E discrete time, (19) b t Ft[θ]= F[θ] , (16) and the distribution of the transient capacity is expressed as b b h(θ)(J )e−yl h(−θ)(J )e−yu where F[θ] = F1[θ] is a E × E matrix with ijth element 1 − 0 ≤ P C(t) ≤ c∗ ≤ 0 , (ij) θx (ij) (θ) (−θ) F [θ] = pij e F (dx), and θ ∈ Θ = {θ ∈ R : min(h (Jj )) min(h (Jj )) θx bij b j∈E j∈E e F ( )(dx) 0 for the lower bound. (i)b The theorem indicates that the distribution of wireless where K[θ]=Λ+ κ (θ) diag +λij qij Bij [θ] − 1 [16]. By channel capacity with Markov dependence is light-tailed. Perron-Frobenius theorem, F[θ] has a positiveb real eigenvalue with maximal absolute value, eκ(θ), in discrete time; K[θ] Analytical and computational results of transient capacity are b has a real eigenvalue with the maximal real part, κ(θ), in shown in Fig. 1. continuous time. The corresponding right and left eigenvectors III. DEPENDENCE CONTROL (θ) (θ) (θ) (θ) are respectively h = hi and v = vi , A. Performance Measure i∈E i∈E particularly, v(θ), v(θ)h(θ) =1 and πh(θ) =1, where π = v(0) The wireless channel is essentially a queueing system with is the stationary distribution and h(0) = e. cumulative service process S(t) and cumulative arrival process t A(0,t)= a(s), where a(t) denotes the traffic input to the Proof. Consider the delay-constrained capacity for the con- s=0 stant fluid process A(t)= λt, channel atP time t, and the temporal increment in the system is expressed as C(d, ǫ) = sup λ, (32) X(t)= a(t) − C(t). (21) P (D(t)>d)≤ǫ,∀t the result follows directly from Theorem 2. The queueing principle of the wireless channel is expressed through the backlog in the system, which is a reflected process Remark 1. It’s a folk law that the regularity of arrival or of the temporal increment X(t) [16], i.e., service processes results in better performance measures, and it’s been proved that for some involved system the queue length B(t + 1) = [B(t)+ X(t)]+ , (22) of a constant fluid input is the shortest for all types of inputs assume B(0) = 0, the backlog function is then expressed as that have the same average traffic rate [20], thus the minimal delay and maximal delay-constrained capacity. B(t) = sup (A(s,t) − S(s,t)). (23) 0≤s≤t B. Dependence Classification For a lossless system, the output is the difference between the We classify the dependence into three types, i.e., positive ∗ input and backlog, A (t)= A(t) − B(t), i.e., dependence, independence, and negative dependence. Intu- A∗(t)= A ⊗ S(t), (24) itively, positive dependence implies that large or small values of random variables tend to occur together, while negative where f ⊗ g(t) = inf0≤s≤t{f(s)+ g(s,t)} is the bivariate dependence implies that large values of one variable tend to min-plus convolution [17], and the delay is defined via the occur together with small values of others [21]. input-output relationship [18], i.e., We use discrete-time setting in this subsection. Formally, the D(t) = inf {d ≥ 0 : A(t − d) ≤ A∗(t)} , (25) cumulative capacity S is said to have a positive dependence structure S+ in the sense of increasing convex order, if which is the virtual delay that a hypothetical arrival has expe- rienced on departure. The maximum rate of traffic with delay S⊥ ≤icx S+, (33) requirement that the system can support without dropping is or a negative dependence structure S− in the sense of increas- defined as the delay-constrained capacity or throughput [19]: ing convex order, if A(t) C(d, ǫ) = sup E . (26) S− ≤icx S⊥, (34) P (D(t)>d)≤ǫ,∀t t where S⊥ has an independence structure. Since the mean of The above results also apply to continuous-time setting. sum of random variables equals the sum of means of individual To embody the impact of dependence in wireless channels, random variables, i.e., we assume that the input is a constant fluid process, i.e., E[S−]= E[S⊥]= E[S+], (35) A(t)= λt. (27) the increasing convex ordering and convex ordering of cumu- The results of performance metrics are summarized in the lative capacity are equivalent [22], i.e., following theorem and corollary. Theorem 2. Consider a constant arrival process A(t) = λt, S− ≤icx S⊥ ≤icx S+ ⇐⇒ S− ≤cx S⊥ ≤cx S+. (36) the delay conditional on the initial state J0 = i is bounded by It’s worth noting that the supermodular ordering of instanta- h(−θ)(J )e−θλd h(−θ)(J )e−θλd neous capacity, i.e., i i (28) (−θ) ≤ Pi(D ≥ d) ≤ (−θ) , C ≤sm C, (37) max h (Jj ) min h (Jj ) j∈E j∈E indicates that the marginal distributionse of the instantaneous where −θ is the negative root of κ(θ)=0 of S(t) − λt and increments are identical, particularly, if C ≤sm C, then (−θ) h is the corresponding right eigenvector, given the initial n C(i) ≤ n C(i). i=1 cx i=1 e state distribution ̟, the delay and backlog are bounded by As the distribution of wireless channel capacity is light- P P e tailed, the asymptotic behavior of the bounding function is P (D ≥ d) = ̟iPi(D ≥ d), (29) Xi∈E still exponential for weak forms of dependence and becomes P (B ≥ b) = P (D ≥ b/λ). (30) heavy-tailed for stronger dependence [15]. The ordering of the exponential adjustment coefficient is as follows. Corollary 1. For constant fluid traffic A(t) = λt, the delay- constrained capacity, letting P (D ≥ d) = ǫ, is expressed as Theorem 3. Consider two wireless channel capacity processes, if the cumulative capacities are convex ordered, then (−θ) (−θ) ǫ · max h (Jj ) ǫ · min h (Jj ) the adjustment coefficients for the delay bounds are corre- −1 j∈E −1 j∈E spondingly ordered, i.e., log (−θ) ≤ λ ≤ log (−θ) . θd ̟ih (Ji) θd ̟ih (Ji) i∈E i∈E N P P (31) S(t) ≤cx S(t), ∀t ∈ =⇒ θ ≤ θ. (38) e e Proof. Consider the negative increment process, i.e., −X(t)= C(t) − a(t). (39) If it is light-tailed, then the delay violation probability has an exponential bound with adjustment coefficient θ > 0 defined by κ(θ)=0, where [15], [23]

1 θ t X(i) κ(θ) = lim E e Pi=1 . (40) t→∞ t h i By exploring the ordering of the cumulative increment process, n n −X(i) ≤cx −X(i), ∀n ∈ N, (41) Xi=1 Xi=1 e the adjustment coefficients are ordered as follows [15], [23] θ ≤ θ. (42) Specifically, for constant arrival,e the ordering of the cumulative capacity results in the ordering of the cumulative negative Fig. 2. Delay tail distribution of Rayleigh channel. “-” and “+” depict increment process. respectively negative and positive dependence in capacity, the lines depict the double-sided bounds with the intervals depicted as the shaded areas. The ordering of the adjustment coefficients gives an or- λ = 10kbits, W = 20kHz, SNR = e0.5 for the independence case dering of the asymptotic delay tail distribution, with some of additive capacity process, SNR = [e0.5 e0.5; 0.7e0.5 0.7e0.5], and restrictions, the result can be applied to the delay-constrained P = [0.4125 0.5875; 0.2518 0.7482] calculated from Fréchet copula with α = 0.5 for λ − C(t) indicating negative dependence in capacity and capacity. P = [0.2875 0.7125; 0.3054 0.6946] calculated from Fréchet copula with α = −0.5 for λ − C(t) indicating positive dependence in capacity, for the Corollary 2. For delay bounding functions with the same dependence case of Markov additive capacity process with initial distribution prefactor or are bounded by a same prefactor before the ̟ = [0.5 0.5] and stationary distribution π = [0.3 0.7]. exponential term, the ordering of the cumulative capacity S− ≤cx S⊥ ≤cx S+ indicates the ordering of the delay, i.e., and no further information is provided by the lagged values P (D− ≥ x) ≤ P (D⊥ ≥ x) ≤ P (D+ ≥ x) , (43) of other variables [8]. and the ordering of the delay-constrained capacity for the Proposition 2. For a n-dimensional process X, same prefactor, i.e., X1,..., Xi−1, Xi+1,..., Xn do not Granger cause Xi, if [8], [24] λ− ≥ λ⊥ ≥ λ+. (44) 1 n i i F X ,...,X i F X (45) The impact of negative dependence and positive dependence P Xtk+1 ≤ x| tk = P Xtk+1 ≤ x| tk . in capacity on delay and comparison with independence in No-Granger causality and Markov property of each process capacity are illustrated in Fig. 2. We fix the noise power with respect to its natural filtration together imply the Markov density and change the transmission power in SNR. The result structure of the system as a whole [8], [24]. Additional restric- shows that the wireless channel attains a better performance tion is required for the converse to hold, the 2-dimensional with less transmission power or smaller capacity for negative result is available in [8], and the following theorem is an dependence in contrast to positive dependence. extension to n-dimensional case. C. Dependence under Control Theorem 4. For a n-dimensional Markov process X consist- We distinguish the random parameters in the wireless sys- ing two dimension sets X and X, X = X ∪ X, X does not tem, which cause the dependence in the wireless channel ca- Granger cause X, if and only if pacity, into two categories, i.e., uncontrollable parameters and controllable parameters. Uncontrollable parameters represent C u , u , u , 1 j,j+1 Xj Xj Xj+1 uX the property of the environment that can not be interfered, j+1 X X C j (u j ) e.g., fading, while controllable parameters represent the con- u u (46) = CX X ⋆ CXj Xj+1 X , Xj+1 , figurable property of the wireless system, e.g., power. We use j j j the controllable parameters to induce negative dependence into X does not Granger cause X, if and only if the wireless channel capacity to achieve dependence control.

We assume no Granger causality among random parameters. Cj,j uX , u , 1uX , u +1 j Xj j+1 Xj+1 No-Granger causality is a concept initially introduced in CX uX econometrics and refers to a multivariate dynamic system in j j u u (47) = CX X ⋆ CX X Xj , X . which each variable is determined by its own lagged values j j j j+1 j+1 Remark 2. Speciﬁcally, for the wireless channel capacity Algorithm 1 Algorithm for Dependence Control that is modeled by a multivariate Markov process, let X and Model: A n-dimensional Markov process consisting of n 1- X represent respectively the uncontrollable and controllable dimensional Markov processes without Granger causality parameters. The no-Granger causality guarantees that if the Result: Transition matrix of the controllable parameter uncontrollable and controllable parameters form a multivari- 1: Initialisation: Cj,j+1, Cj , and ̟ ate Markov process, the processes of the uncontrollable and 2: for j =0 to t − 1 do controllable parameters are also Markov processes, which is 3: for 1 ≤ i ≤ n of interest do i i i i necessary in dependence control because we need to model 4: Calculate Pj and ̟j+1 = ̟j Pj , with i i i the uncontrollable parameters with a certain process and to ̟j Pj = Cj,j+1 conﬁgure the controllable parameters in a certain way based 5: endP for on a certain process. 6: end for 7: return P A stronger restriction is that all the 1-dimensional Markov processes do not Granger cause each other, and the results are as follows. where x and y are the ordered state space vector, the state

Theorem 5. For a n-dimensional Markov process X with distribution at j is ̟j = ̟ 0≤k≤j Pk, and Fj (sj ) = temporal copula Cj,j+1 and spatial copula Cj , ̟j (sk ≤ sj) and Fj+1 = ̟Qj Pj . Together with the unity propertyP of transition matrix j∈E pij = 1, ∀i ∈ E, the i X1 Xn i Xi (48) P Xtk+1 ≤ x| tk ,..., tk = P Xtk+1 ≤ x| tk , transition probabilities Pj areP obtained. if and only if Proof. For random variables X and Y with the copula C [5]

1 n i E (IY

C (F (0), F (0)) = π0p00, (57a) Cj,j+1(u1, v1,u2, 1) = CX2,X1 ⋆ CX1,X1 (v1,u1,u2), (50) j j j j+1  C (F (1), F (0)) = π0p00 + π1p10, (57b) 1 2  and X does not Granger cause X , if and only if [8]  C (F (1), F (1)) = π0 (p00 + p01)+ π1 (p10 + p11) , (57c) Cj,j+1(u1, v1, 1, v2)= CX1,X2 ⋆ CX2,X2 (u1, v1,u2). (51) C (F (0), F (1)) = π (p + p ) . (57d) j j j j+1  0 00 01   In the special case, if the spatial dependence is expressed by Given a stationary distribution [π0 π1], F (0) = π0 and the product copula, then F (1) = π0 + π1, we obtain the values of p00 and p10 from the equations, and we further obtain p01 = 1 − p00 and Cj,j+1(u1, v1,u2, 1) = v1CX1X1 (u1,u2) , (52) j j+1 p11 =1 − p10 from the unity property. Cj,j+1(u1, v1, 1, v2) = u1CX2X2 (v1, v2) . (53) j j+1 The algorithm of dependence control is shown in Algorithm Since the copula requires continuity by deﬁnition, interpola- 1. It’s worth noting that the Markov property is a pure tion is needed to construct a copula from the transition matrix property of copula, different copula functions provide a way of a Markov process [5], while it’s not needed to calculate to character the negative or positive dependence, based on the transition matrix from a copula. The approach to calculate which we can calculate the transition matrix of the controllable the transition probability of a Markov chain given the copula parameters in the wireless system, e.g., power, and bring their of the two consecutive levels is summarized in the following impacts into capacity. The Markov family copula is elaborated theorem. in Appendix A-A. An example is illustrated in Fig. 3. The fading process is Theorem 6. For a 1-dimensional Markov process with ﬁnite independent and the power changes with negative or positive state space E and initial distribution ̟, given the copula dependence, the result shows that the times series of the between successive levels C , j,j+1 instantaneous capacity exhibits weakly negative dependence

̟j (sj )Pj (sj ,sj+1 ≤ y)= Cj,j+1 (Fj (x), Fj+1(y)) , or weakly positive dependence, and the impact is manifested

sXj ≤x in the transient capacity, on the other hand, it indicates that (54) the impact of independent parameters is strong. a better performance when the capacity bears negative dependence relative to positive dependence. Numerical results showed that a better performance is attained even with a smaller average capacity. 3) We distinguished the random parameters in the wireless system into two categories, i.e., uncontrollable parameters and controllable parameters. We assumed that these two types of parameters do not Granger cause each other, for which we provided a sufficient and necessary condition based on the copula of the Markov process. We proposed to use the controllable parameters to induce negative dependence into the capacity, specifically, we constructed a sequence of temporal copulas of the Markov process based on a priori information of the random parameter processes, from which we calculated the transition matrix of the controllable parameters, given the expected dependence information. Thus, we achieved dependence control. It’s worth noting that though this paper focuses on dependence with Markov property, the influence of positive and negative dependence holds in general.

APPENDIX A Fig. 3. Wireless channel capacity of Markov additive Rayleigh channel. The COPULA uncontrollable parameter is fading with one state and the controllable parameter is power with two states. The Markov process is time homogeneous without Consider a joint distribution F (X ,...,Xn) with marginal Granger causality. The dependence structure is Gaussian copula with correla- 1 tion matrix Σ = [10.1 −0.5 0; 0.1100; −0.5010.1; 0 0 0.1 1] as negative distribution Fi(x), i =1,...,n. Denote ui = Fi (Xi), which dependence (left column), Σ = [10.1 0.5 0; 0.1 1 0 0; 0.5010.1; 0 0 0.1 1] is uniformly distributed in the unit interval, then [25] as positive dependence (right column), initial distribution ̟ = [0.5 0.5], stationary distribution π = [0.3 0.7]. W = 20kHz and SNR = −1 −1 F (X1,...,Xn)= F F (u1) ,...,F (un) (58) [e0.5 e0.5; 0.7e0.5 0.7e0.5]. 1000 time slots. Correlation coefﬁcient and 1 n probability value between the time series and lag-1 series are provided. ≡ C (u1,...,un) , (59)

where C is a copula with standard uniform marginals, specif- IV. CONCLUSION ically, if the marginals are continuous, the copula is unique. We discovered a hidden resource in wireless channels, Definition 1. A n-dimensional copula C is a distribution func- namely stochastic dependence. Specifically, when the wireless tion on [0, 1]n with standard uniform marginal distributions, channel capacity evolves with negative dependence, a better if channel performance is attained with a smaller capacity. The 1) C(u ,...,u ) is increasing in each component u ; contributions of this paper are as follows. 1 n i 2) C(1,..., 1,ui, 1,..., 1) = ui for all i ∈{1,...,n}, ui ∈ 1) We modeled the wireless channel capacity as a Markov [0, 1]; additive process, used copula to represent the dependence n 3) For all (a1,...,an), (b1,...,bn) ∈ [0, 1] with ai ≤ bi, in the underlying Markov process, and derived double- sided distribution function bounds of the cumulative 2 2 i1+...+in capacity and transient capacity. Both continuous-time ... (−1) C(u1,i1 ,...,un,in ) ≥ 0, and discrete-time setting were considered. We treated iX1=1 iXn=1 the wireless channel as a queueing system and derived where uj,1 = aj and uj,2 = bj for all j ∈{1,...,n}. tail bounds of delay and backlog for a constant fluid arrival process, which is usually regarded as the best Example 4. The extremely positive dependence, indepen- performance measure that the wireless channel attains. dence, and extremely negative dependence are expressed by In addition, we obtained a double-sided bound of delay- copulas. For 2-dimensional copula, the extremely positive cop- constrained capacity as the inverse function of delay. ula, product copula (independence), and extremely negative 2) We classified the dependence in wireless channel ca- copula are defined as pacity into three types by defining stochastic orders on M(x, y) = min(x, y), (60) the capacity, namely positive dependence, independence, and negative dependence. In terms of the performance P (x, y) = xy, (61) measure, we proved that the wireless channel attains W (x, y) = max(x + y − 1, 0). (62) For a n-dimensional copula function C, the extremely positive if and only if [5], [7], for s 0. (68) if α is near −1, strongly negative dependence is indicated. In case the copula C is symmetric, i.e., C(x) = C(1 − x), It’s elaborated that Fréchet copulas imply quite a restricted the mapping is expressed as T (x)=1 − x. type of Markov process and Archimedean copulas are incom- patible with Markov chains [26]. The Sklar’s theorem depicts that every distribution can be written as a copula function taking marginals as arguments, APPENDIX B and every copula function taking arbitrary marginals as ar- PROOF OF THEOREM 1 guments is a joint distribution. In addition, the functional invariance property implicates that the dependence structure Proof. In order to provide exponential upper bound for the represented by a copula is invariant under non-decreasing and distribution of the cumulative capacity, define [27] continuous transformations of the marginals. min (h(θ)(J )) j∈E j θS(t)−tκ(θ) (78) L(t)= (θ) e , A. Markov Family Copula h (J0) To construct a Markov process, the copula must satisfy the where L(t) ≤ L(t), i.e., E[L(t)] ≤ 1. Apply Markov Markov property condition [5], [7]. inequality to L(t) and get, for any µ> 0, Definition 2. A family of 2n-dimensional copula Cst, s

Csu(1, ·) = Cut(·, 1), (69) (θ) −tκ(θ)+θx minj∈E (h (Jj )) Cu Choose µ = e h(θ)(J ) , for θ ≤ 0, Cst = Csu ⋆ Cut, (70) 0 (θ) for all . h (J0) tκ θ θx s 0, Example 5. The n-dimensional Gausssian copula is expressed h(θ)(J ) 0 tκ(θ)−θx (81) as P {S(t) ≥ x}≤ (θ) e , minj∈E (h (Jj )) −1 −1 CΣ(u)=ΦΣ Φ (u1),..., Φ (un) , (71) which indicates that the distribution has a light tail. Letting ∗ where ΦΣ denotes the joint distribution of the n-dimensional −y = tκ(θ) − θx ≤ 0, the distribution of the transient standard normal distribution with linear correlation matrix Σ, capacity is bounded by and Φ−1 denotes the inverse of the distribution function of the h(θ)(J )e−yl h(θ)(J )e−yu 1-dimensional standard normal distribution. 0 ∗ 0 1 − (θ) ≤ P C(t) ≤ c ≤ (θ) , min(h (Jj )) min(h (Jj )) Example 6. A convex combination of M, P , and W is a j∈E j∈E ∗ (82) Markov family copula, i.e., ∗ tκ(θ)+y ∗ where c = θt , with y = yu for θ < 0 for the upper ∗ Cst = α(s,t)W + (1 − α(s,t) − β(s,t))P + β(s,t)M, (72) bound, and y = yl for θ> 0 for the lower bound. APPENDIX C REFERENCES PROOF OF THEOREM 2 [1] E. C. Niehenke, “Wireless communications: Present and future,” IEEE Microwave Magazine, vol. 15, no. 2, pp. 26–35, 2014. Proof. For the constant fluid arrival A(t) = λt, the delay [2] J. Hecht et al., “The bandwidth bottleneck,” Nature, vol. 536, no. 7615, is expressed as P (D(t) > x) = P {sup {λ(t − s) − pp. 139–142, 2016. 0≤s≤t [3] D. Tse and P. Viswanath, Fundamentals of wireless communication. S(s,t)} > λx}, and the relationship with P (B(t) > x) Cambridge university press, 2005. x [4] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. Soong, directly follows, i.e., P (B(t) > x)= P D(t) > λ . and J. C. 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