Wireless Channel Capacity Analysis of Cumulative Capacity Analysis of Maximum and Minimum Cumulative Capacity Further Properties of Wireless Channel Capacity Fengyou Sun and Yuming Jiang Norwegian University of Science and Technology (NTNU) Third Workshop on Network Calculus, April 06, 2016, M¨ unster Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity Analysis of Cumulative Capacity Analysis of Maximum and Minimum Cumulative Capacity Outline 1 Wireless Channel Capacity Background Motivation 2 Analysis of Cumulative Capacity General Results Special Cases 3 Analysis of Maximum and Minimum Cumulative Capacity General Results Special Cases Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity Background Analysis of Cumulative Capacity Motivation Analysis of Maximum and Minimum Cumulative Capacity Instantaneous Capacity Wireless fading channels are time variant and wireless channel capacity is a stochastic process [Tse, 2005] The instantaneous capacity of the channel at time t can be expressed as a function of the instantaneous SNR γ t at this time [Costa and Haykin, 2010] C ( t ) = log 2 ( g ( γ t )) Statistical properties of first order and second order have been investigated [Rafiq, 2011, P¨ atzold, 2011] mean, variance, PDF, CDF, LCR, and ADF Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity Background Analysis of Cumulative Capacity Motivation Analysis of Maximum and Minimum Cumulative Capacity Motivation of this Work Capacity and QoS requirements in future wireless communication more data (500 EB), higher data rate (1000 × , 100 × ), and less latency ( < 1ms, round-trip) in 5G [Andrews et al., 2014] Instantaneous capacity is not sufficient for use in assessing if data transmission over the channel meets its QoS requirements capacity behavior of average sense ergodic capacity temporal behavior of the capacity LCR, ADF Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity Background Analysis of Cumulative Capacity Motivation Analysis of Maximum and Minimum Cumulative Capacity Fundamental Concepts Cumulative capacity t � S ( s , t ) ≡ C ( i ) i = s +1 Maximum cumulative capacity k � S (0 , t ) ≡ sup S ( j , k ) = sup C ( i ) 1 ≤ j ≤ k ≤ t 1 ≤ j ≤ k ≤ t i = j forward-looking and backward-looking variations − → S (0 , k ) , ← − S (0 , t ) ≡ sup S (0 , t ) ≡ sup S ( j , t ) 1 ≤ k ≤ t 1 ≤ j ≤ t Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity Background Analysis of Cumulative Capacity Motivation Analysis of Maximum and Minimum Cumulative Capacity Fundamental Concepts (Cont’d) Minimum cumulative capacity k � S (0 , t ) ≡ 1 ≤ j ≤ k ≤ t S ( j , k ) = inf inf C ( i ) 1 ≤ j ≤ k ≤ t i = j forward-looking and backward-looking variations → (0 , t ) ≡ S 1 ≤ k ≤ t S (0 , k ) , S inf − (0 , t ) ≡ 1 ≤ j ≤ t S ( j , t ) inf − ← Range of cumulative capacity R (0 , t ) ≡ S (0 , t ) − S (0 , t ) Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity General Results Analysis of Cumulative Capacity Special Cases Analysis of Maximum and Minimum Cumulative Capacity Exact Expression The CDF of the cumulative capacity is expressed as � F S ( s , t ) ( x ) = dF H ( h s +1 , h s +2 , . . . , h t ) , S ( s , t )= � t i = s +1 log 2 (1+ γ | h i | 2 ) ≤ x where F H ( h s +1 , h s +2 , . . . , h t ) is the joint distribution of channel gains, e.g. the multivariate generalized Rician distribution [Beaulieu and Hemachandra, 2011] � ∞ m − 1 √ t 2 S m − 1 exp( − ( t + S 2 )) I m − 1 (2 S F H ( h 1 , h 2 , . . . , h N ) = t ) t =0 � √ t σ 2 k λ 2 N � , h k k dt . 1 − Q m Ω k Ω k k =1 Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity General Results Analysis of Cumulative Capacity Special Cases Analysis of Maximum and Minimum Cumulative Capacity Standard Bounds The CDF of the cumulative capacity satisfies the following inequalities: F l S ( s , t ) ( r ) ≤ F S ( s , t ) ( r ) ≤ F u S ( s , t ) ( r ) , where t � F u S ( s , t ) ( r ) ≡ inf F C ( i ) ( r i ) , � t i = s +1 r i = r 1 i = s +1 + t � F l S ( s , t ) ( r ) ≡ sup F C ( i ) ( r i ) − ( t − s − 1) . � t i = s +1 r i = r i = s +1 Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity General Results Analysis of Cumulative Capacity Special Cases Analysis of Maximum and Minimum Cumulative Capacity Improved Bounds Let F 1 = . . . = F n =: F be distribution functions on R + . Then for any s ≥ 0 it holds that [Puccetti and R¨ uschendorf, 2012] � � � s − ( n − 1) u n F ( t ) dt M + u n ( s ) ≤ D ( s ) = u < s / n min inf , 1 , s − nu � � � s − ( n − 1) u F ( t ) dt n m + u n ( s ) ≥ d ( s ) = sup max − n + 1 , 0 , s − nu u > s / n where � � n � � � M + n ( t ) = sup P X i ≥ t ; X i ∼ F i , 1 ≤ i ≤ n , i =1 � � n � � � m + n ( t ) = inf P X i > t ; X i ∼ F i , 1 ≤ i ≤ n . i =1 Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity General Results Analysis of Cumulative Capacity Special Cases Analysis of Maximum and Minimum Cumulative Capacity Comonotonicity The set A ⊆ R n is said to be comonotonic if for any x ≤ y or y ≤ x holds, where x ≤ y denotes the componentwise order, i.e., x i ≤ y i for all i = 1 , 2 , . . . , n . [Dhaene et al., 2002] In the special case that all marginal distribution functions are identical F C ( i ) ∼ F C , comonotonicity of C ( i ) is equivalent to saying that C ( s + 1) = C ( s + 2) , . . . , = C ( t ) holds almost surely [Dhaene et al., 2002], i.e., � � x F S ( s , t ) ( x ) = F C . t − s Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity General Results Analysis of Cumulative Capacity Special Cases Analysis of Maximum and Minimum Cumulative Capacity Independence If C ( i ) and C ( j ), i � = j , are independent, f S ( s , t ) = f C ( s +1) ∗ . . . ∗ f C ( t ) , where ∗ � x denotes the convolution operation, namely, F S ( s , t ) ( x ) = −∞ f S ( s , t ) ( y ) dy . According to the central limit theorem, F S ( s , t ) ( x ) approaches a normal distribution [Papoulis and Pillai, 2002], i.e., � x − E [ S ( s , t )] � F S ( s , t ) ( x ) ≈ G . σ 2 [ S ( s , t )] For identical marginals F C ( i ) ∼ F C , according to the Markov inequality P { L t ≥ µ } ≤ 1 µ E [ L t ] = 1 µ, P { S t ≥ x } ≤ e θ x − t κ ( θ ) , � e θ x F ( dx ), L t = e θ S t − t κ ( θ ) , and L t is a mean-one where κ ( θ ) = log E e θ C ( i ) = log martingale [Asmussen, 2003]. Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity General Results Analysis of Cumulative Capacity Special Cases Analysis of Maximum and Minimum Cumulative Capacity Markov Process For a Markov additive process, denote matrix � F [ θ ] with ij the element � e θ x F ( ij ) ( dx ), where F ij ( dx ) = P i , 0 ( J 1 = j , Y 1 ∈ dx ), Y n = S n − S n − 1 . � F ( ij ) [ θ ] =: By Perron-Frobenius theory, the matrix � F [ θ ] has a positive real eigenvalue with maximal absolute value e κ ( θ ) and the corresponding right eigenvector h ( θ ) = ( h ( θ ) F [ θ ] h ( θ ) = e κ ( θ ) h ( θ ) . [Asmussen, 2003] ) i ∈ E , i.e., � i Let L n = h ( θ ) ( J n ) h ( θ ) ( J 0 ) e − θ S n + n κ ( θ ) , L n = min n ( h ( θ ) ( J n )) e − θ S n + n κ ( θ ) , h ( θ ) ( J 0 ) according to Markov inequality [Gallager, 2013] µ E [ L n ] ≤ 1 1 P { L n ≥ µ } ≤ µ, e − n κ ( θ )+ θα h ( θ ) ( J 0 ) / min n ( h ( θ ) ( J n )) . P { S n ≥ α } ≤ Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
Wireless Channel Capacity General Results Analysis of Cumulative Capacity Special Cases Analysis of Maximum and Minimum Cumulative Capacity Non-Granger Causality Assumption Non-Granger causality refers to a multivariate dynamic system in which each variable is determined by its own lagged values and no further information is provided by the lagged values of the other variables. Then the copula function representing the dependence structure among the running maxima (minima) at time t n is the same copula function (survival copula function) representing dependence among the levels at the same time [Cherubini and Romagnoli, 2010]. Fengyou Sun and Yuming Jiang, NTNU Further Properties of Wireless Channel Capacity
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