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Low-Latency Millimeter-Wave Communications: Traffic Dispersion or Network Densification?
arXiv:1709.08410v1 [cs.NI] 25 Sep 2017
Guang Yang, Ming Xiao, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Abstract—Low latency is critical for many applications in wireless communications, e.g., vehicle-to-vehicle (V2V), multimedia, and industrial control networks. Meanwhile, for the capability of providing multi-gigabits per second (Gbps) rates, millimeter-wave (mmWave) communication has attracted substantial research interest recently. This paper investigates two strategies to reduce the communication delay in future wireless networks: traffic dispersion and network densification. A hybrid scheme that combines these two strategies is also considered. The probabilistic delay and effective capacity are used to evaluate performance. For probabilistic delay, the violation probability of delay, i.e., the probability that the delay exceeds a given tolerance level, is characterized in terms of upper bounds, which are derived by applying stochastic network calculus theory. In addition, to characterize the maximum affordable arrival traffic for mmWave systems, the effective capacity, i.e., the service capability with a given quality-of-service (QoS) requirement, is studied. The derived bounds on the probabilistic delay and effective capacity are validated through simulations. These numerical results show that, for a given average system gain, traffic dispersion, network densification, and the hybrid scheme exhibit different potentials to reduce the end-to-end communication delay. For instance, traffic dispersion outperforms network densification, given high average system gain and arrival rate, while it could be the worst option, otherwise. Furthermore, it is revealed that, increasing the number of independent paths and/or relay density is always beneficial, while the performance gain is related to the arrival rate and average system gain, jointly. Therefore, a proper transmission scheme should be selected to optimize the delay performance, according to the given conditions on arrival traffic and system service capability. Index Terms—Millimeter-wave, delay, traffic dispersion, network densification.
I. I NTRODUCTION A. Background Wireless communications in millimeter-wave (mmWave) bands (from around 24 GHz to 300 GHz) is a key enabler for multi-gigabits per second (Gbps) transmission [1]–[3]. In contrast to conventional wireless communications in sub6 GHz bands, many appealing properties, including the abundant spectral resources, lower component costs, and highly directional antennas, make mmWave communications attractive for future mobile communications standards, e.g., ECMA, This work was supported by EU Marie Curie Project, QUICK, No. 612652, and Wireless@KTH Seed Project “Millimeter Wave for Ultra-Reliable LowLatency Communications”. Guang Yang and Ming Xiao are with the Department of Information Science and Engineering (ISE), KTH Royal Institute of Technology, Stockholm, Sweden (Email: {gy,mingx}@kth.se). H. Vincent Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ, USA (Email:
[email protected]).
IEEE 802.15.3 Task Group 3c (TG3c), IEEE 802.11ad standardization task group, and Wireess Gigabit Alliance (WiGig). As an important metric for evaluating the quality of service (QoS), low latency plays a crucial role in the forthcoming fifth generation (5G) mobile communications [4]–[6], especially for various delay-sensitive applications, e.g., high-definition television (HDTV), intelligent transport system, vehicle-toeverything (V2X), machine-to-machine (M2M) communication, and real-time remote control. The overall delay in wireless communications consists of four components as follows [7], [8]: propagation delay (time for sending the one bit to its designated end via the physical medium), transmission delay (time for pushing the packet into the communication medium in use), processing delay (time for analyzing a packet header and making a routing decision), and queuing delay (the time that a packet spends in the buffer or queue, i.e., waiting for transmission). Normally, the overall delay for queuing system is dominantly determined by the queuing delay, while the contributions by the other types of delay are nearly negligible. Thus, for low-latency buffer-aided systems, the major task is to largely decrease the queuing delay. In recent years, many efforts from different aspects have been devoted to low-latency mmWave communications. In [4], several critical challenges and possible solutions for delivering end-to-end low-latency services in mmWave cellular systems were comprehensively reviewed, from the perspectives of protocols at the medium access control (MAC) layer, congestion control, and core network architecture. By applying the Lyapunov technique for the utility-delay control, the problem of ultra-reliable and low-latency in mmWave-enabled massive multiple-input multiple-output (MIMO) networks was studied in [9]. Regarding hybrid beamforming in mmWave MIMO systems, a novel algorithm for achieving the ultralow latency of mmWave communications was proposed in [10], where the training time can be significantly reduced by progressive channel estimations. Furthermore, for systems with buffers at transceivers, the probabilistic delay for pointto-point mmWave communications is analyzed in [11], where the delay bound is derived based on network calculus theory. B. Motivation Due to unprecedented data volumes in mmWave communications, the transceivers for many applications are commonly equipped with large-size buffers, such that the data arrivals that cannot be processed in time will be temporarily queued up in the buffer until corresponding service is provided. Hence, the low-latency problem for mmWave communications with
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buffers can be interpreted as a delay problem in queuing systems, equivalently. By queuing theory, it is known that the key idea for effectively reducing the queuing delay is to keep lower service utilization. That is, the average arrival rate of data traffic should be less than the service rate of server as much as possible. Commonly, low service utilization can be fulfilled mainly through two distinct methods: offloading arrival traffic and improving the service capability. In a wireless network, offloading arrival traffic can be realized by adopting the traffic dispersion scheme, and service enhancement can be realized by adopting the network densification scheme. Traffic dispersion stems from the application of distributed antenna systems (DASs) or distributed remote radio heads (RRHs) in mmWave communications, and network densification is motivated from the trend of dense deployment for mmWave networks. Roughly, the traffic dispersion scheme applies the “divide-and-conquer” principle, which enables parallel transmissions to fully exploit the spatial diversity, such that a large single queue (or large delay, equivalently) can be avoided. On the other hand, the network densification scheme departs from reducing the path loss, via shortening the separation distance between adjacent nodes, such that the end-to-end service capability can be improved. Clearly, both the traffic dispersion and network densification schemes are promising and competitive candidates for lowlatency mmWave communications. Despite that there are many research contributions in low-latency communications based on above two principles, the existing literature focus on either the dispersion scheme (e.g., [12]–[15]) or multi-hop relaying scheme (e.g., [16]–[19]). It is not clear yet which scheme can provide better delay performance. For designing or implementing mmWave networks, it is essential to explore the respective strengths of traffic dispersion and network densification, and know the their applicability and capability for realizing low-latency mmWave networks. Moreover, a combination of traffic dispersion and network densification, termed as “hybrid scheme”, is worthy of study. Intuitively, the hybrid strategy takes advantages of both traffic dispersion and network densification, and potentially can improve the delay performance under certain constraints. C. Objectives and Contributions The main objective of our work is to investigate the potential of delay performance of three transmission schemes for mmWave communications, i.e., traffic dispersion, network densification, and the hybrid scheme. Considering buffer-aided mmWave systems, the delay performance is studied in terms of probabilistic delay and effective capacity, respectively. More precisely, to characterize the violation probability of delay, we derive corresponding probabilistic delay bounds by applying stochastic network calculus based on moment generating function (MGF). Furthermore, effective capacity is investigated to characterize the maximum asymptotic service capability. The main contributions can be summarized as follows: • To the best of our knowledge, the comparison between traffic dispersion and network densification has not been
•
•
•
performed previously, and thus the advantage of each scheme for delay performance is not yet clear. In this paper, we comprehensively investigate the respective strengths of above two strategies. Furthermore, we propose a generic hybrid scheme, i.e., a flexible combination of traffic dispersion and network densification, and study its potentials in reducing the communication delay. Thus, our work not only exhibits the benefits of two basic transmission strategies in different scenarios, but also explores the capability of the novel hybrid scheme in reducing the delay for certain scenarios. Using MGF-based stochastic network calculus, probabilistic delay bounds for traffic dispersion, network densification, and the hybrid scheme are derived, respectively. Compared to most of existing results that only consider homogeneous networks, we study the probabilistic delay heterogeneous settings for the sake of generality. Our work contributes to stochastic network calculus theory by extending the application of stochastic network calculus from homogeneous cases to heterogeneous scenarios. It is known that, effective capacity can be used to analyze the maximum service capability in the asymptotic sense. Despite that significant progresses have been achieved in recent years, there however still remain several open issues for effective capacity, e.g., generic formulations for considered transmission schemes and analysis for networks with an arbitrary number of tandem servers. In our work, in addition to evaluating effective capacity for three schemes, we also provide closed-form lower bounds according to the specific fading characteristics of mmWave channels, which enable us to keep the track of the asymptotic service capability in a more convenient manner. It is demonstrated that, traffic dispersion, network densification and the hybrid scheme have respective advantages, and resulting end-to-end delay performance depends on the average system gain (involving the system configuration, sum power constraint, and end-toend communication distance) and the density of data traffic (e.g., average arrival rate). For instance, when the given average system gain is large, traffic dispersion, the hybrid scheme, and network densification are suitable for the scenarios with heavy, medium, and light arrival traffic, respectively. However, when the given average system gain is small, the corresponding strengths of above three schemes significantly change with respect to arrival traffic. These observations provide interesting insights for mmWave network designs and implementations. That is, the transmission scheme for low-latency performance should be properly selected according to the density of arrival traffic and/or the feasible system gain.
The remainder of this paper is outlined as follows. In Sec. II, preliminaries for MGF-based stochastic network calculus are provided. In Sec. III, we give system models for traffic dispersion, network densification, and the hybrid scheme, respectively, and present MGF-based characterizations for traffic and service processes in mmWave systems. For three low-
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latency schemes, corresponding probabilistic delay bounds are derived in Sec. IV, by applying MGF-based stochastic network calculus. In Sec. V, the effective capacity and its lower bound for three schemes are analyzed. Performance evaluations are presented in Sec. VI, where the derived results are validated, and the delay performance of three schemes is discussed. Conclusions are finally drawn in Sec. VII.
In this section, for illustration purpose, we will review network calculus theory briefly and present preliminaries for MGF-based stochastic network calculus. More details for the presented fundamental results can be found in [20]–[25]. A. Traffic and Service Process Considering a fluid-flow, discrete-time queuing system with a buffer of infinite size, within time interval [s, t), 0 ≤ s ≤ t, the non-decreasing bivariate processes A(s, t), D(s, t) and S(s, t) are defined as the cumulative arrival traffic to, departure traffic from, and service offered by server, respectively. We assume A (s, t), D (s, t) and S (s, t) are stationary non-negative random processes, and their values are zeros whenever s = t. Furthermore, the cumulative arrival and service processes are the sum of instantaneous realizations of each time slot within the given time interval. More exactly, let ai and si represent the instantaneous values of arrival and service in the ith time slot, respectively, then A(s, t) and S(s, t) are given as t−1 X
ai and S(s, t) =
t−1 X
si ,
(1)
i=s
i=s
for all 0 ≤ s ≤ t, where time slots are normalized to 1 time unit. For network calculus, the convolution and deconvolution in (min, +)-algebra, denoted by ⊗ and , respectively, are two critical operators for deriving performance bounds of queuing systems. Their definitions with respect to the non-decreasing and strictly positive bivariate processes X(s, t) and Y (s, t) are respectively given as (X ⊗ Y )(s, t) , inf {X(s, τ ) + Y (τ, t)} and
(3)
B. MGF-based Probabilistic Bounds For queuing systems with stochastic traffic and/or service processes, the MGF-based stochastic network calculus [25] is used to effectively characterize the probabilistic delay. Among various MGF-based approaches for stochastic processes, the Chernoff bound is widely used in stochastic network calculus for deriving probabilistic bounds. More exactly, for random variable X and given x, the Chernoff bound on P (X ≥ x) is given as P (X ≥ x) ≤ e−θx E eθX = e−θx MX (θ),
where E [Y ] and MY (θ) are the mean value and the MGF (or the Laplace transform) with respect to Y , respectively, and θ is a positive free parameter. For any stochastic process X(s, t), t ≥ s, the MGF of X for any θ ≥ 0 is defined as [26] h i MX (θ, s, t) , E eθX(s,t) . Moreover, MX (θ, s, t) , MX (−θ, s, t) = E e−θX(s,t) is also defined, likewise. The following two inequalities regarding the MGF, associated with convolution ⊗ and deconvolution , respectively, are extensively applied in MGF-based network calculus [26]. More exactly, let X (s, t) and Y (s, t) be independent random processes, we have MX⊗Y (θ, s, t) ≤ and MX Y (θ, s, t) ≤
s≤τ ≤t
(X Y )(s, t) , sup {X(τ, t) − Y (τ, s)}. 0≤τ ≤s
With respect to cumulative arrival A(s, t) and cumulative service process S (s, t), according to network calculus, cumulative departure D(s, t) is characterized as [22] D(s, t) ≥ (A ⊗ S)(s, t),
where the equality can be achieved if the system is linear [21]. Then, in terms of A (s, t) and D (s, t), the virtual delay at time t is defined as W (t) , inf{w ≥ 0 : A(0, t) ≤ D(0, t + w)}, which is further upper bounded by W (t) ≤ inf{w ≥ 0 : (A S)(t + w, t) ≤ 0}.
Snet (s, t) = (S1 ⊗ S2 ⊗ · · · ⊗ Sn ) (s, t) ,
where Si (s, t) for any 1 ≤ i ≤ n denotes the service process on the ith hop.
II. P RELIMINARIES FOR N ETWORK C ALCULUS
A(s, t) =
Moreover, an appealing and important property of network calculus theory is the capability of dealing with tandem systems, where the equivalent end-to-end network service process can be computed as the (min, +) convolution of the individual service processes. More exactly, given n concatenated servers, the end-to-end network service process Snet (s, t) is given by
(2)
t X
MX (θ, s, u) · MY (θ, u, t)
(4)
MX (θ, u, t) · MY (θ, u, s) .
(5)
u=s
s X
u=0
Based on (4) and (5), many properties for MGF-based stochastic network calculus are summarized in [26]. For the tandem network shown in (3), the corresponding MGF is written as MSnet (θ, s, t) for the system with n tandem servers, which is bounded by MSnet (θ, s, t) ,MS1 ⊗S2 ⊗···⊗Sn (θ, s, t) n X Y ≤ MSi (θ, ui−1 , ui ) , (6) s≤u1 ≤···≤un−1 ≤t i=1
where u0 = s and uN = t, and Si , i = 1, . . . N denotes the service process on each hop. (6) is obtained via applying the union bound and independence assumption [26].
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Fig. 2. Hybrid scheme for low-latency mmWave communications.
(a) Traffic Dispersion
•
(b) Network Densification Fig. 1. Illustrations of two schemes for reducing latency for mmWave communications: (a) traffic dispersion, and (b) network densification.
Assuming independent arrival traffic and service process, in terms of MGF-based characterizations for cumulative arrival traffic and cumulative service process, i.e., MA (θ, s, t) and MS (θ, s, t), respectively, we define MA,S (θ, s, t) as min(s,t)
MA,S (θ, s, t) ,
X
u=0
MA (θ, u, t) · MS (θ, u, s).
(7)
Then, based on (7), the violation probability is bounded as P (W (t) ≥ w) ≤ P ((A S)(t + w, t) ≥ 0) ≤ inf MA S (θ, t + w, t) θ>0
(8)
≤ inf MA,S (θ, t + w, t) , w , θ>0
where the Chernoff bound and the inequality in (5) are used, and w denotes the violation probability of delay. Solving w, we can obtain [19], [26] w = inf w ˜ ≥ 0 : inf {MA,S (θ, t + w, ˜ t)} ≤ w . (9) θ>0
III. S YSTEM D ESCRIPTIONS AND MGF- BASED T RAFFIC /S ERVICE C HARACTERIZATIONS A. System Model Two schemes, i.e., traffic dispersion and network densification, are illustrated in Fig. 1. They work as follows: • For traffic dispersion (as shown in Fig. 1a), the original arrival traffic is firstly partitioned into multiple substreams by the data splitter. Then, each sub-stream gets served and delivered towards the receiver through the given path, independently. Finally, the receiver combines all sub-streams through the data merger from different paths, thereby forming the output traffic. Thanks to the application of beamforming techniques with narrow beams or highly directional antennas in mmWave communications, the co-channel interference is negligible, and hence it is reasonable to assume independence for multiple sub-streams and propagation paths. The principle of traffic dispersion is to decompose the original heavy
arrival traffic into multiple lighter ones, thereby to avoid a large queue in the buffer. For network densification (as shown in Fig. 1b), multiple relaying nodes 1 as servers are deployed along the sourcedestination transmission path. Due to the concatenation of relying nodes, the output traffic from one relay can be treated as the input traffic for the subsequent connected relay. The application of multi-hop relaying follows trend of ultra-dense mmWave networks. Likewise, it is feasible to assume independent channel conditions on multiple hops, thanks to high directivity and propagation properties for mmWave communications. In contrast to the principle of traffic dispersion, the mechanism of network densification is deploying a large number of relaying nodes between the given source and destination, which can reduce the path loss on each hop via shortening the separation distance between adjacent nodes, thereby increase the end-to-end service capability.
Besides, combining the benefits of traffic dispersion and network densification, we consider hybrid scheme as shown in Fig. 2. The original arrival traffic is firstly divided into multiple sub-streams by data splitter. Subsequently, these sub-streams are allocated with independent paths for data transmission, and each path consists of multiple relaying nodes. It is evident that, this combination takes advantages of traffic dispersion and network densification, i.e., offloading the arrival traffic and enhancing the service capability. For the propagation characteristic, it is known from [2], [27] that, shadowing is the dominant channel fading effect in mmWave bands, which is commonly characterized by a log-normally distributed random variable. To be more precise, given separation distance l and path loss exponent α, the capacity of the mmWave channel is given as C = B log2 1 + κξγl−α , (10)
where γ denotes the transmit power normalized by the background noise, ξ (dB) ∼ N 0, ν 2 corresponds to the lognormal shadowing effect with variance ν 2 , B denotes the bandwidth, and κ is a scalar depending on system configurations, i.e., antenna gains of the communication pair and employed radio frequencies. 1 Full-duplex relaying nodes are used throughout this paper, where the self interference is ignored for simplifying analysis.
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B. MGF Bounds for Service Processes According to (8), it can be found that, MA (θ, s, t) and MS (θ, s, t) are required to compute the probabilistic delay bound. For MA (θ, s, t), we consider the (σ (θ) , ρ (θ)) characterization 2 for arrival traffic introduced by [20], [22], where MA (θ, s, t) is upper bounded by MA (θ, s, t) ≤ eθ(ρ(θ)(t−s)+σ(θ)) , eθσ(θ) µt−s (θ) ,
(11)
for ∀θ > 0. Moreover, for cumulative service process S (s, t), in terms of channel capacity by (10), we have S (s, t) =
t−1 X k=s
C (k) = η
t−1 X k=s
ln 1 + κξ (k) γl−α ,
where η , B log2 e, and C (k) denotes the instantaneous channel capacity with respect to shadowing effect ξ (k) at time k. Then, the corresponding MGF can be written as −ηθ t−s (k) −α MS (θ, s, t) = E 1 + κξ γl , (12)
where i.i.d. shadowing effect across the time dimension is assumed. However, it is intractable to derive an exact expression of (12) for the specific fading characteristic of mmWave channels, since the shifted log-normal random variable has no closedform inverse moment. Instead, we apply a parameterized upper bound proposed in [11], to characterize the inverse moment. More exactly, in terms of parameter δ > 0, which balances computational complexity and bound tightness, there exists a parameterized upper bound Uδ,C (θ), such that E e−θC ≤ Uδ,C (θ) holds for any θ > 0. After that, MS (θ, s, t) in (12) can be upper bounded as MS (θ, s, t) ≤
t−s Uδ,C
(ηθ) ,
(13)
with Uδ,C (θ) for any θ > 0 given as Nδ (u) X −θ aθ,δ (k)FΞ (kδ) , Uδ,C (θ) = min (1 + δNδ (u)) + u≥0 k=1
where FΞ (∗) denotes the cumulative distribution function (c.d.f.) of random variable Ξ , κξγl−α , which follows the log-normal distribution assumed in this paper. In addition, we −θ −θ let aθ,δ (k) = (1 + (k − 1)δ) − (1 + kδ) , and Nδ (u) = u b δ c. One can refer to [11] for more details. Note that, with stochastic arrival traffic and/or service processes, it is intractable to obtain the closed-form probability distribution function (p.d.f.) for delay. Besides, the MGF-based approach gives bounds, instead of the actual delay. Therefore, for schemes to be investigated subsequently, it is infeasible and meaningless to formulate optimization problems with certain given constraints to optimize the actual delay performance. In this sense, , our work mainly aims to characterize the delay performance via bounds, rather than to optimize the schemes. 2 This (σ (θ) , ρ (θ)) arrival process covers various traffic models, e.g., the exponentially bounded burstiness (EBB), effective bandwidth arrival characterization and deterministically bounded arrivals.
IV. P ROBABILISTIC D ELAY B OUNDS FOR L OW-L ATENCY T RANSMISSION S CHEMES In this section, we mainly focus on the performance analysis of traffic dispersion and network densification, and derive upper bounds for probabilistic delay. Subsequently, based on the derived results for above two basic schemes, the delay bound for the hybrid scheme is also presented. A. Delay Bound for Traffic Dispersion As shown in Fig. 1a, assuming n ≥ 1 independent paths for the traffic dispersion scheme, the original cumulative arrival A (s, t) is decomposed into several sub-streams Ai (s, t) for 1 ≤ i ≤ n, i.e., A (s, t) =
n X
Ai (s, t) .
i=1
For each sub arrival traffic Ai (s, t), we assume that its MGF also follows (σi (θ) , ρi (θ)) characterization, i.e., MAi (s, t) ≤ eθ(ρi (θ)(t−s)+σi (θ)) , eθσi (θ) µt−s (θ) . i
(14)
In this sense, with respect to (σ (θ) , ρ (θ)) characterization for A (s, t), we have ρ (θ) =
n X
ρi (θ) and σ (θ) =
i=1
n X
σi (θ) .
i=1
Moreover, for each transmission path, the channel capacity of the ith path at time k is given as 0(k) (k) Ci = B log2 1 + κξi γi l−α , (k)
where separation distance l is assumed for each path, and ξi and γi denote the instantaneous shadowing effect and normalized transmit power on the ith path, respectively. Based on Pt−1 0(k) (13), the upper bound for the MGF of Si0 (s, t) , k=s Ci , with respect to any given δ > 0, can be written as t−s t−s MSi0 (θ, s, t) ≤ Uδ,C (θ) . 0 (θ) , ψi i
(15)
Without loss of generality, considering heterogeneous traffic dispersion, i.e., σi (θ) 6= σj (θ), µi (θ) 6= µj (θ) or ψi (θ) 6= ψj (θ) may hold for 1 ≤ i 6= j ≤ n, an upper bound on the violation probability for traffic dispersion is given in the following theorem. Theorem 1. Let W (t) , max {W1 (t) , W2 (t) , · · · , Wn (t)} be the delay for the traffic dispersion scheme with n independent paths, where Wi (t) denotes the delay on the ith path. Then, for any w ≥ 0, the probabilistic delay is bounded as follows: + θσi (θi ) w n Y e ψi (θi ) P (W (t) ≥ w) ≤ 1 − 1 − inf , θi >0 1 − µi (θi ) ψi (θi ) i=1 whenever the stability condition µi (θi ) ψi (θi ) < 1 holds for + some θi > 0 and all 1 ≤ i ≤ n, where [x] , max {x, 0} for x ∈ R. Proof: Please refer to Appendix A.
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In Theorem 1, the stability condition, i.e., µi (θi ) ψi (θi ) < 1 for all 1 ≤ i ≤ n, should be satisfied to obtain the above probabilistic delay bound. This stability condition stems from the fact that, to avoid infinite delay, the arrival rate of each sub-stream cannot exceed the service capability provided on the corresponding path. Furthermore, Theorem 1 tells that, the path with higher service utilization, i.e., higher µi (θi ) ψi (θi ), is the main contributor to large delay. With homogeneous settings, i.e., σi (θi ) = σ (θ), µi (θ) = µ (θ) and ψi (θ) = ψ (θ) for all 1 ≤ i ≤ n, we have the following corollary. Corollary 1.1. For homogeneous traffic dispersion, given any w ≥ 0, the probabilistic delay bound of Theorem 1 is given as " n #+ θσ(θ) w e ψ (θ) , P (W (t) ≥ w) ≤ 1 − 1 − inf θ>0 1 − µ (θ) ψ (θ) whenever µ (θ) ψ (θ) < 1 holds for some θ > 0. B. Delay Bound for Network Densification
MA (s, t) ≤ eθσ(θ) µt−s (θ) . Moreover, for the n-hop relaying channel, we denote by (k) 00(k) Ci = B log2 1 + κξi γi li−α
the instantaneous channel capacity of the ith hop at time k, (k) where ξi , γi and li denote the instantaneous shadowing effect, normalized transmit power, and transmission distance on the ith hop, respectively. Likewise, based on (13), given certain δ > 0, the upper bound for the MGF of Si00 (s, t) , Pt−1 00(k) can be written as k=s Ci t−s t−s MSi00 (θ, s, t) ≤ Uδ,C (θ) . 00 (θ) , φi
(16)
i
Considering heterogeneous multi-hop relaying, i.e., φi (θ) 6= φj (θ) may hold for 1 ≤ i 6= j ≤ n, we have the following theorem regarding the probabilistic end-to-end delay bound. Theorem 2. Let W (t) be the end-to-end delay of an n-hop network. Then, for any w ≥ 0, the probabilistic delay is bounded as follows:
n X Y
n P
i=1
πi =v
i=1
(µ (θ) φi (θ))
Corollary 2.1. For homogeneous network densification, given any w ≥ 0, the probabilistic delay bound of Theorem 2 is given as θσ(θ) e · B (θ; w, n) ,1 , P (W (t) ≥ w) ≤ min inf θ>0 µw (θ)
whenever the stability condition µ (θ) φ (θ) < 1 holds for some θ > 0. Here, B (θ; w, n) is given as B (θ; w, n) , min {B1 (θ; w, n) , B2 (θ; w, n)} ,
with B1 (θ; w, n) and B2 (θ; w, n) respectively defined as w n−1+w (µ (θ) φ (θ)) B1 (θ; τ, n) , n n−1 (1 − µ (θ) φ (θ))
and
In the network densification scheme for multi-hop mmWave networks, we again assume that A (s, t) follows the (σ (θ) , ρ (θ)) characterization, i.e.,
P (W (t) ≥ w) θσ(θ) ∞ X e ≤ inf w θ>0 µ (θ) v=w
provided by each hop. Besides, by Theorem 2, the hop with higher service utilization, i.e., higher µ (θ) φi (θ), is the main contributor to a large delay. In the following corollary, the probabilistic delay for multihop relaying with homogeneous settings is studied, where φi (θ) = φ (θ) for all 1 ≤ i ≤ n are assumed.
πi
B2 (θ; τ, n) ,
1 n (1 − µ (θ) φ (θ)) n+w−1 w−1 − (µ (θ) φ (θ)) . n
Proof: Please refer to Appendix C.
C. Delay Bound for the Hybrid Scheme In the hybrid scheme with m (m ≥ 1) independent transmission paths, for the arrival traffic, we assume that the sub-stream Ai (s, t) follows from (σ (θ) , ρ (θ)) characterization given in (14). Furthermore, we denoted by (k) (k) −α Ci,j = B log2 1 + κξi,j γi,j li,j ,
the instantaneous channel capacity of the j th hop on the ith (k) transmission path at time k, where ξi,j , γi,j and li,j represent the instantaneous log-normal shadowing effect, normalized transmit power and propagation distance, respectively. Then, similarly, according to (13), the upper bound for the MGF of Pt−1 (k) Si,j (s, t) , k=s Ci,j with respect to certain δ > 0 can be written as t−s MSi,j (θ, s, t) ≤ Uδ,C (θ) , ϕt−s i,j (θ) . i,j
(17)
In light of above, the probabilistic delay bound for the hybrid scheme is presented in the following theorem. ,
whenever the stability condition µ (θ) φi (θ) < 1 holds for some θ > 0 and all 1 ≤ i ≤ n.
Proof: Please refer to Appendix B. Similarly, the stability condition indicates that, to avoid infinite delay, the arrival rate cannot exceed the service capability
Theorem 3. Assuming m (m ≥ 1) independent paths for the hybrid scheme system, with ki hops on the ith path for 1 ≤ i ≤ m, we define pˆi for any given w ≥ 0 as km ∞ eθσi (θi ) X X Y πj pˆi , inf (µ (θ ) ϕ (θ )) . i i i,j i w θi >0 µi (θi ) v=w kP j=1 m πj =v j=1
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Then, the end-to-end probabilistic delay is upper bounded as P (W (t) ≥ w) ≤ 1 −
m Y
i=1
+
[1 − pˆi ] ,
whenever the stability condition µi (θi ) ϕi,j (θi ) < 1 holds for some θi > 0 and all 1 ≤ i ≤ m and 1 ≤ j ≤ ki .
Proof: Applying Theorem 2 in Theorem 1 for each independent path, it is then straightforward to conclude the probabilistic delay bound for the hybrid scheme. Particularly, with homogeneous settings in the hybrid scheme, i.e., σi,j (θ) = σ (θ), µi,j (θ) = µ (θ), ϕi,j (θ) = ϕ (θ), and ki = k for all 1 ≤ i ≤ m and 1 ≤ j ≤ ki , the result by Theorem 3 can be further reduced, which can be presented via combining Corollary 1.1 and Corollary 2.1. For brevity, results related to the homogeneous scenario are omitted. V. E FFECTIVE C APACITY A NALYSIS WITH G IVEN AVERAGE S YSTEM G AIN From the analysis of Sec. IV that, it is important to ensure that the arrival rate is below the service capability (refer to the stability conditions for three schemes). That is, the service capability characterizes the limiting potentials to deal with data traffic without causing infinite delays. Thus, in what follows, we use effective capacity [28], which is another important metric related to the delay performance departing from asymptotic service capability, to analyze three schemes. A. Basics for Effective Capacity In light of the MGF of the service process by (12), the effective capacity is defined as log MS (θ, 0, t) (18) , −θt where θ > 0 represents the QoS exponent, which indicates a more stringent QoS requirement for a higher θ. Assuming that the queue is not empty. That is, the system always attempts to transmit. Then the violation probability of delay can be equivalently approximated as [29], [30] C (−θ) , lim
t→∞
P (W (t) ≥ wmax ) ≈ exp (−C (−θ) wmax ) ,
(19)
whenever
log P (W (t) ≥ x) = −C (−θ) , x where wmax denotes the maximum system tolerance of delay. In addition, for the stability consideration of first-in-first-out (FIFO) queuing systems in the asymptotic sense, according to the G¨artner-Ellis Theorem [31], the arrival and service process should satisfy the following condition with given θ, i.e., lim
x→∞
log MA (θ, 0, t) ≤ R∗ (θ) , C (−θ) , (20) θt where R (θ) in terms of QoS exponent θ is commonly termed as the “effective bandwidth” [22], and the maximum effective bandwidth, denoted by R∗ (θ) is characterized by the effective capacity C (−θ). We notice that, the relation between effective R (θ) , lim
t→∞
bandwidth and effective capacity, i.e., R (θ) ≤ C (−θ), follows the intuition of asymptotic stability (or stability in the longterm sense), and its principle resembles the stability condition stated in Theorem 1, Theorem 2, or Theorem 3. According to the properties above, it is clear that, for any given QoS exponent θ > 0, a larger effective capacity C (−θ) not only indicates a stronger capability for serving heavier arrival traffic (see (20)), but also leads to faster decay in the probability delay (refer to (19)). Therefore, in the following analysis, we focus on the effective capacity (or the maximum effective bandwidth, equivalently) of traffic dispersion, network densification and the hybrid scheme, respectively. In what follows, for fair comparison and convenient analysis, homogeneous settings are assumed for traffic dispersion, network densification, and the hybrid scheme, with respect to given end-to-end communication distance and sum power constraint for all transceivers. More exactly, let L be the source-destination distance, and consider following homogeneous settings for three transmission schemes, respectively: • For the traffic dispersion scheme, given n independent paths, we assume that γi = n−1 γ for all 1 ≤ i ≤ n. • For the relaying densification scheme, given n hops, we assume that γi = n−1 γ and li = n−1 L for all 1 ≤ i ≤ n. • For the hybrid scheme, given m ≥ 1 independent paths and k ≥ 1 relaying nodes per path, such that m · k = n (m or k is a divisor of n, equivalently), we assume that γi,j = n−1 γ and li,j = k −1 L for all 1 ≤ i ≤ m and 1 ≤ j ≤ k. According to settings above, it is easy to find that, traffic dispersion and network densification can be treated as two extreme cases of the hybrid scheme, i.e., corresponding to m = n and m = 1, respectively. For notional simplicity, we denote the system gain in the subsequent derivations by Ξ , κξγL−α , which is a log-normally distributed random variable due to random variable ξ. Note that, Ξ in dB (with respect to ν in dB) can be rewritten as (dB)
Ξ(dB) , E [Ξ]
+ ν · X,
(21)
(dB)
where X ∼ N (0, 1), and E [Ξ] = 10 log10 (κγL−α ) is termed as the average system gain. Besides, we define T as (dB) ην 2 θ E [Ξ] − , 10 log10 e 200 log210 e , (22) T , η · max (dB) E [Ξ] ην 2 θ − + log 2 20 log10 e 800 log210 e
which will be used to derive lower bounds for the effective capacity in following analysis.
B. Effective Capacity of Traffic Dispersion For traffic dispersion, the effective capacity of each independent path can be obtain as 1 CSi0 (−θ) = − log E [exp (−θCi0 )] , θ
8
where the independence of channel condition across the time dimension is applied. Then, considering n parallel independent paths, the effective capacity of traffic dispersion is given as CS 0 (−θ)
,CPn
0 i=1 Si
= − lim
t→∞
(−θ) =
n X i=1
n X i=1
MS 00 (θ, 0, t) ≤
X
Pn
i=1
CSi0 (−θ)
log MSi0 (θ, 0, t) . θt
(23)
Based on (23) and homogeneous settings for traffic dispersion, the maximum effective bandwidth for traffic dispersion is obtained as h −ηθ i n . R∗S 0 (θ) = − · log E 1 + n−1 Ξ θ With the log-normal fading characteristic of the mmWave channel, the lower bound for R∗S 0 (θ) is given in Theorem 4. Theorem 4. For homogeneous traffic dispersion with n independent paths, given θ > 0, the maximum effective bandwidth is lower bounded as +
R∗S 0 (θ) ≥ n · [T − log (n)] . Proof: Please refer to Appendix D. Due to the asymptotic tightness of the lower bound (easy to be obtained from the derivation), we can approximately have R∗S 0
since
(θ) ≈ n · (T − log (n)) ,
whenever T is sufficiently large, e.g., high average system gain E [Ξ]. As we can see, the multiplexing gain is roughly n, which indicates significant improvement in effective capacity by increasing the number of independent paths.
n h Y −ηθ iπi E 1 + κξγi li−α
πi =t i=1
h −ηθ it = E 1 + nα−1 Ξ ·
X
Pn
i=1
1
πi =t
−ηθ it t+n−1 h = E 1 + nα−1 Ξ , n−1
where the second line is achieved due to the uniformly allocated transmit power (normalized) and the identical length of each hop. Therefore, the lower bound for R∗S 00 (θ) can be obtained as h −ηθ it α−1 E log t+n−1 1 + n Ξ n−1 R∗S 00 (θ) ≥ lim t→∞ −θt h −ηθ i n−1 α−1 log E 1 + n Ξ log (t+n−1) (n−1)! ≥ − lim t→∞ −θ θt h i −ηθ α−1 log E 1 + n Ξ = , −θ where the inequality in the second line uses the property that N Nk , ≤ k! k
for any integers N ≥ k ≥ 0. Likewise, based on the log-normal fading characteristic of the mmWave channel, the lower bound for R∗S 00 (θ) is given in Theorem 5. Theorem 5. For homogeneous network densification with n independent hops, given θ > 0, the maximum effective bandwidth is lower bounded as +
R∗S 00 (θ) ≥ [T + (α − 1) log (n)] . Proof: Please refer to Appendix E. Likewise, according to the asymptotic tightness of the lower bound, we can approximately have
C. Effective Capacity of Network Densification For network densification, by (3), the network service process for the multi-hop relying scheme is characterized by S 00 (0, t) , (S100 ⊗ S200 ⊗ · · · ⊗ Sn00 ) (0, t) .
According the definition of effective capacity, we have CS 00 (−θ) , − lim
t→∞
log MS100 ⊗S200 ⊗···⊗Sn00 (θ, 0, t) . θt
(24)
Note that (min, +) convolution is involved for characterizing the concatenated service in network densification, it is intractable to derive the closed-form expression for effective capacity. Based on (24) and the homogeneous settings, the maximum effective bandwidth for the network densification scheme is lower bounded by h −ηθ i 1 , R∗S 00 (θ) , CS 00 (−θ) ≥ − log E 1 + nα−1 Ξ θ
R∗S 00 (θ) ≈ T + (α − 1) log (n) ,
whenever T is sufficiently large. We can see that, the approximation of R∗S 00 (θ) increases with the logarithm of n, while the multiplexing gain is 1. Thus, in contrast to traffic dispersion, the improvement of effective capacity by increasing the relay density is much smaller. Moreover, it is easy to verify that, for large T and given n, we have T + (α − 1) log (n) n · (T − log (n)) .
In this sense, network densification is definitely inferior to traffic dispersion. D. Effective Capacity of the Hybrid Scheme The effective capacity of the hybrid scheme is obtained as m X log MSi,1 ⊗···⊗Si,ki (θ, 0, t) , t→∞ θt i=1
CS (−θ) = − lim
(25)
9
Theorem 6. For the homogeneous hybrid scheme with m independent paths and n/m relaying nodes per path, given θ > 0, the maximum effective bandwidth is lower bounded as R∗S
+
(θ) ≥ m · [T + (α − 1) log (n) − α log (m)] .
Proof: The theorem is immediately concluded by straightforwardly applying the variable substitution for Theorem 4 or Theorem 5. Similarly, thanks to the asymptotic tightness of the lower bound, when T is sufficiently large, the approximation of R∗S (θ) can be written as R∗S (θ) ≈ m · (T + (α − 1) log (n) − α log (m)) .
For m = n and m = 1, it is easy to find that, R∗S (θ) can be reduced to R∗S 0 (θ) and R∗S 00 (θ), respectively. Furthermore, when T is sufficiently large, it is easy to show that R∗S (θ) is monotonically increasing with respect to m (according to the partial derivative over m). Thus, given the sufficiently large average system gain, we have R∗S 00 (θ) < R∗S (θ) < R∗S 0 (θ) .
(26)
VI. P ERFORMANCE E VALUATION In this section, we provide numerical results for the probabilistic end-to-end delay bound and effective capacity discussed in Sec. III and IV, respectively. Through simulations, we firstly validate the derived bounds for probabilistic delay and effective capacity, where the respective advantages of traffic dispersion and network densification are evaluated and discussed 3 . Subsequently, for the hybrid scheme (including traffic dispersing and network densification), the factors and conditions for achieving low-latency mmWave communications are extensively studied. For fairness considerations, the homogeneous settings presumed in Sec. IV are applied. The general system configurations are summarized as follows: the bandwidth is allocated with B = 500 MHz, the path loss exponent α = 2.45, the standard deviation of shadowing effect is given as ν = 8 dB, and δ = 10−3 is adopted for the precision parameter in (13). 3 Two extreme cases, i.e., traffic dispersion and network densification, are considered only for simplifying comparison, and comprehensive results regarding the generic hybrid scheme are presented afterwards.
10-1
Violation Probability ǫw
where m is a divisor of n, denoting the number of independent paths in the hybrid scheme. Similarly, applying the log-normal fading characteristic of the mmWave channel, the lower bound for R∗S (θ) is presented in the following theorem.
100
n=2
10-2
10-3
n = 10
10-4
n=5 10-5
10-6
Simulated Upper Bound 0
1
2
3
4
5
Delay w Fig. 3. Violation probability w v.s. targeted delay bound w for the traffic dispersion scheme, where ρ = 2 Gbps and E [Ξ] = 13.4 dB.
100 Simulated Upper Bound
10-1
Violation Probability ǫw
where we assume m independent paths and ki relaying nodes on the ith path (1 ≤ i ≤ m). According to (25) and the homogeneous settings for the hybrid scheme, in light of the above derivations for traffic dispersion and network densification, the maximum effective bandwidth for the hybrid scheme is lower bounded by " −ηθ # nα−1 m ∗ , Ξ RS (θ) ≥ − · log E 1 + θ mα
n=2
10-2
10-3
10-4
n = 10 n=5
10-5
10-6
0
1
2
3
4
5
Delay w Fig. 4. Violation probability w v.s. targeted delay bound w, for the multi-hop relaying scheme, where ρ = 2 Gbps and E [Ξ] = 13.4 dB.
Simulations are realized via MATLAB, and the number of iterations for generating stochastic instances is 107 . A. Bound Validation The violation probabilities of delay for traffic dispersion and network densification are illustrated in Fig. 3 and Fig. 4, respectively, where average system gain is E [Ξ] = 13.4 dB and the arrival rate is ρ = 2 Gbps. In Fig. 3, it can be seen that, for all cases, i.e., n = 2, 5 and 10, the probabilistic delay bound derived in Corollary 1.1 accurately characterizes the slope of the simulated result. We notice that, although the violation probability of delay decreases as the number of independent paths grows, i.e., n increasing from 2 to 10, the resulting performance improvement is not remarkable under the given average system gain and arrival rate. Furthermore, in Fig. 4, likewise, the bound by Corollary 2.1 is also able to well predict the decaying rate of violation probability. However, in
10
70
14 Sim.: n=5 L.B.:n=5 Sim.: n=15 L.B.: n=15
50
12
Network Densification
4
40 2 30 0 0
1
E[Ξ] = E[Ξ] = E[Ξ] = E[Ξ] = E[Ξ] =
Traffic Dispersion
Effective Capacity [Gbps]
Effective Capacity [Gbps]
60
2
20
10
10
0 dB 5 dB 10 dB 15 dB 20 dB
Traffic Dispersion
Network Densification
8
6
4
2
0
0 0
10
20
30
40
50
Average System Gain E[Ξ] [dB] Fig. 5. Effective Capacity C (θ) v.s. average system gain E [Ξ] for two data transmission schemes, where QoS exponent θ = 2.
contrast to the traffic dispersion scheme, increasing the relay density (equivalently, increasing the number of relays) can significantly decrease the probabilistic delay. Fig. 5 illustrates the effective capacity for traffic dispersion and network densification, with respect to different average system gain E [Ξ] and given QoS exponent θ = 2. Clearly, the derived lower bounds of effective capacity, i.e., Theorem 4 and Theorem 5 for traffic dispersion and network densification, respectively, are asymptotically tight when the average system gain goes to sufficiently large. We notice that, when the average system gain is high, e.g., E [Ξ] ≥ 15 dB, traffic dispersion exhibits remarkable advantages as n grows. For instance, at E [Ξ] = 20 dB, the effective capacity by traffic dispersion dramatically boosts from around 7 Gbps to 15 Gbps when n increases from 5 to 15, while by network densification it only slightly grows from around 4 Gbps to 5 Gbps. However, for lower average system gain, i.e., E [Ξ] ≤ 2 dB, the network densification scheme adversely outperforms its counterpart, although the relative advantage is limited. The findings above indicate that, the benefit of traffic dispersion diminishes as the average system gain or the number of independent paths decreases. In this sense, network densification can be a better option in cases of sparse relay deployment and low average system gain, which are in line with the results demonstrated by Fig. 3 and Fig. 4. B. Simulation and Discussion Including two extreme cases, i.e., traffic dispersion (m = n) and network densification (m = 1), the effective capacity of the hybrid scheme with 1 ≤ m ≤ n for given n = 12 is illustrated in Fig. 6, where the QoS exponent is θ = 2, and average system gain E [Ξ] varies from 0 to 20 dB. It can be seen that, for the lower average system gain, e.g., E [Ξ] = 0 dB, if increasing the number of independent paths, the effective capacity firstly grows and then decays for m ≥ 3. In this case, to largely reduce the end-to-end
0
2
4
6
8
10
12
14
nubmer of paths m
Fig. 6. Effective capacity C (−θ) v.s. number of independent paths m for the hybrid scheme with respect to θ = 2 and n = 12, where the number of independent paths is m = 1, 2, 3, 4, 6 or 12.
delay, the optimal solution is to assign m = 3 independent transmission paths for traffic dispersion, and on each path there should be 4 hops (equivalently, 3 relaying nodes deployed) for network densification. However, as the average system gain rises, the effective capacity tends to be monotonically increasing with respect to the number of independent paths, which is in line with (26). Thus, it is beneficial to apply traffic dispersion if the average system gain is high. The above results demonstrate that, the hybrid scheme with proper configurations, i.e., the proper numbers of independent paths and hops per path, respectively, should be carefully considered to maximize the effective capacity, if the average system gain is small, and the resulting performance can outperform other two counterparts. However, when the average system gain is high, traffic dispersion can provide the best performance. Given violation probability w = 10−3 , the targeted delay tolerances against different arrival rates for three transmission schemes, i.e., traffic dispersion (m = 12), network densification (m = 1) and the hybrid scheme (m = 3), are provided in Fig. 7. For both groups in terms of E [Ξ], clearly, the targeted delay exponentially increases when the arrival rate increases. These drastic growths are resulted by the higher service utilization. That is, the arrival rate approaches the limiting service capability. Besides, for different densities of arrival traffic and average system gains, by comparing the resulting w with respect to any given ρ, it is evident that three transmission schemes exhibit respective strengths. To be more precise, in the group with average system gain E [Ξ] = 5 dB, the traffic dispersion scheme has the largest delay for all rates of the arrival traffic, and the hybrid scheme outperforms the network densification scheme only when the arrival rate is medium or high, e.g., ρ ≥ 2 Gbps. However, if the average system gain is elevated to E [Ξ] = 15 dB, when encountering light (e.g., ρ ≤ 3 Gbps), medium (e.g., 3 ≤ ρ ≤ 7.5 Gbps) and heavy (e.g., ρ ≥ 7.5 Gbps) arrival traffic, respectively,
11
A PPENDIX A P ROOF OF T HEOREM 1
40 Densification (m = 1) Hybrid (m = 3) Dispersion (m = 12)
35 30
Delay w
We start with deriving the upper bound for MAi ,Si0 :
E[Ξ] = 5 dB
25
(a)
MAi ,Si0 (θ, s, t) ≤
E[Ξ] = 15 dB
20 15
min(s,t)
X
eθσi (θ) µt−u (θ) ψis−u (θ) i
u=0
(b) θσi (θ) t−s =e µi
(θ)
10
(c)
≤ eθσi (θ) µt−s (θ) i
5
2
4
6
8
10
Arrival Rate ρ [Gbps]
Fig. 7. Probabilistic delay w v.s. arrival rate ρ for traffic dispersion, network densification and the hybrid scheme, respectively, with respect to violation probability w = 10−3 , where n = 12.
we can see that network densification, the hybrid scheme, and traffic dispersion can produce the smallest end-to-end delays, correspondingly. The above results not only coincide with the conclusions drawn from Fig. 6, but also indicate the importance of considering the arrival rate and the proper transmission scheme jointly to reduce the delay.
(µi (θ) ψi (θ))
v
v=τ
τ
(27)
By the definition of W (t), it is easy to obtain that P (W (t) ≥ w) , P max {Wi (t)} ≥ w 1≤i≤n
(d)
=1 − ≤1 −
(f )
≤1 −
We have considered traffic dispersion and network densification for low-latency mmWave communications, and have investigated their end-to-end delay performance. We have also proposed a hybrid scheme to further reduce latency in certain scenarios. Based on MGF-based stochastic network calculus and effective capacity theory, respectively, we have derived performance bounds for probabilistic delay and effective capacity for the three schemes, which have been validated through simulations. These results have demonstrated that, given the average system gain, traffic dispersion, network densification, and the hybrid scheme show different potential for low-latency mmWave communications. In addition, it has been shown that, increasing the number of independent paths and/or the number of relays for network densification is always advantageous for reducing the end-to-end communication delay, while the performance gain heavily relies on the density of arrival traffic and average system gain, jointly. Thus, it is crucial to select the proper scheme according to the given arrival traffic and system service capability.
v
where τ , max {s − t, 0}. Here, inequality (a) is obtained by plugging (14) and (15) into (7). By performing the change of variable, i.e., v = s−u, equality (b) is achieved. In (c), we let s go to infinity. In the final step, the geometric sum converges only when µi (θ) ψi (θ) < 1 holds for certain θ.
(e)
VII. C ONCLUSIONS
v=τ ∞ X
(µi (θ) ψi (θ))
eθσi (θ) µt−s (θ) (µi (θ) ψi (θ)) i , = 1 − µi (θ) ψi (θ)
0 0
s X
n Y
(1 − P (Wi (t) ≥ w))
i=1 n Y i=1 n Y
i=1
1 − inf MAi ,Si0 (θi , t + w, t) θi >0
1 − inf
θi >0
eθσi (θi ) ψiw (θi ) 1 − µi (θi ) ψi (θi )
,
where the independence assumption among distinct paths is used to derive (d), and inequality (e) applies the (8), and (f ) follows from (27). In addition, we notice that P (W (t) ≥ w) ≤ 1 holds for any w ≥ 0, then the theorem is concluded.
A PPENDIX B P ROOF OF T HEOREM 2
Applying (16) in (6), the MGF of n-hop network service process can be characterized as MS 00 (θ, s, t) ,MS100 ⊗S200 ⊗···⊗Sn00 (θ, s, t) n X Y ≤ φπi i (θ) . Pn
i=1
πi =t−s i=1
12
Regarding the infinite sum in (29), on the one hand, we can obtain that ∞ X n−1+v v (µ (θ) φ (θ)) v v=τ ∞ (a) n − 1 + τ X n−1+v v+τ ≤ (µ (θ) φ (θ)) n−1 v v=0 τ (µ (θ) φ (θ)) (b) n − 1 + τ = n , B1 (θ; τ, n) n−1 (1 − µ (θ) φ (θ))
Then, it is easy to obtain that MA,S 00 (θ, s, t) min(s,t)
X
≤
u=0
n P
πi =s−u
i=1
min(s,t) θσ(θ) t−s
=e
X
eθσ(θ) µt−s (θ)
µ
(θ)
X
=e
µ
(θ)
s X
n P
i=1
≤eθσ(θ) µt−s (θ)
X
τi =s−u
n Y
φπi i (θ)
i=1
n X Y
(µ (θ) φi (θ))
πi
n X Y
(µ (θ) φi (θ))
πi
n v=τ P
∞ X
φπi i (θ)
i=1
µs−u (θ)
u=0
θσ(θ) t−s
n Y
πi =v
where (a) applies the fact n−1+v n−1+v−τ n−1+τ ≤ , v v−τ n−1
i=1
i=1
n v=τ P
πi =v
since
(28)
.
i=1
i=1
Similar to Theorem 1, we notice that the stability condition, i.e., µ (θ) φi (θ) < 1 for all 1 ≤ i ≤ n, needs to be satisfied to guarantee the convergence of (28). Regarding the convergence, it is easy to show that, once the stability condition is met, we have µ (θ) φˆ (θ) < 1 equivalently, where φˆ (θ) , max {φi (θ)} . 1≤i≤n
In this case, we obtain that n X Y
n P
πi =v
πi
(µ (θ) φi (θ))
i=1
≤
X
n P
µ (θ) φˆ (θ)
πi =v
Pni=1 πi
.
i=1
i=1
According to combinatorics properties, we notice that X
n P
i=1
µ (θ) φˆ (θ)
πi =v
∞ X
n P πi i=1
=
∞ X
v=τ
v µ (θ) φˆ (θ)
X
n P
1
πi =v
i=1
v n−1+v µ (θ) φˆ (θ) v v=τ ∞ v −n X n − 1 + v ≤ µ (θ) φˆ (θ) = 1 − µ (θ) φˆ (θ) v v=0 =
Therefore, we can demonstrate that, MA,S 00 (θ, s, t) is convergent if the stability condition holds, since −n MA,S 00 (θ, s, t) ≤ eθσ(θ) µt−s (θ) 1 − µ (θ) φˆ (θ) < ∞. Finally, with respect to (8), the theorem then can be concluded by letting s = t + w. A PPENDIX C P ROOF OF C OROLLARY 2.1 With the homogeneous setting, it is easy to obtain that ∞ eθσ(θ) X n − 1 + v v (µ (θ) φ (θ)) . MA,S 00 (θ, s, t) ≤ s−t µ (θ) v=τ v (29)
u i + k ≤ (i + k − u) 1 + i holds for all i ≥ 1 and all k ≥ u, and (b) is obtained whenever the stability condition µ (θ) φ (θ) < 1 holds for some θ, such that the geometric series converges. On the other hand, we also have ∞ X n−1+v v (µ (θ) φ (θ)) v v=τ τ −1 X 1 n−1+v (c) v = − (µ (θ) φ (θ)) n v (1 − µ (θ) φ (θ)) v=0 τ −1 X (d) 1 n−1+v τ −1 ≤ − (µ (θ) φ (θ)) n v (1 − µ (θ) φ (θ)) v=0 1 n+τ −1 (e) τ −1 = (µ (θ) φ (θ)) n − n (1 − µ (θ) φ (θ)) ,B2 (θ; τ, n) where (c) again utilizes the stability condition to ensure the convergence, (d) is true for any 0 < µ (θ) φ (θ) < 1, and (e) is obtained by applying the combinatorial property that m X n+k n+m+1 = . k n+1 k=0
Thus, in terms of B1 (θ; τ, n) and B2 (θ; τ, n) defined above, we then have ( ) B1 (θ; τ, n) , θσ(θ) t−s MA,S 00 (θ, s, t) ≤ e µ (θ) · min . B2 (θ; τ, n)
Paired with (8) and the upper bound for MA,S 00 (θ, s, t), the theorem can be immediately concluded by letting s = t + w. A PPENDIX D P ROOF OF T HEOREM 4 By the definition of R∗S 0 (θ), we know that h −ηθ i n log E n−1 Ξ R∗S 0 (θ) ≥ −θ n log E Ξ−ηθ + ηθ log n = . −θ
13
For the log-normal distributed random variable Ξ, with the aid of specific reformulation in (21), we have log E Ξ−ηθ ηθ log 10 (dB) E [Ξ] +ν·X = log E exp − 10 2 (dB) ηE [Ξ] log 10 1 ην log 10 · θ2 . =− ·θ+ 10 2 10 Thus, we can obtain that
(dB)
R∗S 0
(θ) ≥ n
ηE [Ξ] 1 − 10 log10 e 2
Note that R∗S 0 (θ) ≥ =
ην 10 log10 e
2
θ − η log n .
−ηθ √ −1 n log E 2 n Ξ
h −θηθ i √ n log E Ξ− 2 + ηθ log 2n
−θ Then, with respect to (21), we can easily obtain that R∗S 0 (θ) ≥n
(dB)
!
ηE [Ξ] 1 − 20 log10 e 2
ην 20 log10 e
2
,
√ ! n θ − η log . 2
Thus, the theorem can be immediately concluded by considering the fact that R∗S 0 (θ) ≥ 0. A PPENDIX E P ROOF OF T HEOREM 5 Following the same approach in Theorem 4, we can further derive the lower bound of RS 00 (θ) as h −ηθ i log E nα−1 Ξ R∗S 00 (θ) ≥ −θ log E Ξ−ηθ − (α − 1) ηθ log n = −θ 2 (dB) ηE [Ξ] θ ην = − + (α − 1) η log n. 10 log10 e 2 10 log10 e Furthermore, we also have −ηθ √ log E 2 nα−1 Ξ R∗S 00 (θ) ≥ h ηθ−θ i α−1 log E Ξ− 2 − ηθ log 2n 2 = , −θ which immediately yields 2 (dB) α−1 ηE [Ξ] θ ην R∗S 00 (θ) ≥ − + η log 2n 2 . 20 log10 e 2 20 log10 e
Therefore, the theorem can be immediately concluded by considering the fact that R∗S 00 (θ) ≥ 0.
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