Transient Analysis for Wireless Networks

Transcription

1 Transient Analysis for Wireless Networks Jaya Prakash Champati, Hussein Al-Zubaidy, James Gross School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm, Sweden School of Information Technology (ITE), Halmstad University, Halmstad, Sweden Abstract In this paper we study the transient behavior of multi-hop wireless networks. Our work is motivated by novel applications from the domain of automation and cyber-physical systems where statistical guarantees need to be provided on a perpacket level. Thus, in order to optimize the network, apart from a stationary understanding of the system dynamics, in particular a short-term understanding of the dynamics is also required. To this end, we derive novel transient performance bounds for wireless networks using stochastic network calculus approach. The transient bounds depend on the initial backlog per node of the multi-hop network as well as the instantaneous channel states. We then compare (numerically) the obtained bounds to state-of-the-art bounds (SOTA) that can be obtained from existing stationary bounds as well as simulation of the network. We show that while SOTA transient and stationary bounds are not able to capture the short-term system dynamics well, our proposed novel bound provides a reasonably tight upper bound of the transient system behavior. We thus propose that the presented expressions can serve as base for future work on transient system optimization. Index Terms Transient analysis; machine type communication; stochastic network calculus; time-critical networks; wireless I. INTRODUCTION With the advent of new applications from the automation domain, it is commonly accepted that wireless networks are facing significant design challenges with respect to new quality-of-service demands. In contrast to wireless networks optimized for human-related applications, there is a much stronger emphasis on strict reliability guarantees with respect to rather short deadlines that need to be fulfilled during the operation of the network - in particular for the so-called mission-critical machine-type applications proposed for 5G cellular networks. For instance, safety-critical automation applications can easily have reliability requirements with respect to acceptable packet error rates in the order of 10 9 [1] while end-to-end delays may not exceed a few milliseconds [2], which is in contrast to the typical latencies of a few hundred milliseconds encountered for example in human-related voice communications. One fundamental challenge in this respect is the development of network models, and subsequently network optimization, that target end-to-end performance over very short time spans. Of particular interest is the end-to-end latency of a given flow transported over a multi-hop route, as it is closely related to the performance and safety of the control application that the flow belongs to. For instance, for an event-based safetycritical control application, a short-term network analysis of a given route may reveal that acceptable safety levels are likely to be breached. This then requires either an instantaneous redistribution of network resources to accommodate the delay and reliability requirements or the shut-down of the control application and hence driving the plant to a fail-safe state. While application layer approaches such as safety layers can achieve the latter given a certain parametrization of the plant dynamics [3], achieving the former rests on the understanding of the short-term stochastic fluctuations of the latency of a given route due to fading channel variability. To this end, two aspects need to be taken into account when attempting a better understanding of the short-term behavior of multihop networks: (i) the instantaneous backlog which influence the end-to-end performance of a newly injected, time critical packet sequence, and (ii) the variability of the wireless service that result in random service increments. In combination, these two effects emphasize the necessity of queuing-theoretic modeling approach with respect to transient characterizations of the latency. However, such non-stationary behaviour of multi-hop wireless networks is not well understood as of today, as it requires the analysis of a wireless networked system in transient state. The literature on transient state network analysis is sparse compared to the queuing-theoretic literature body on stationary/steady state metrics. This is due to the fact that transient state queuing analysis pursued so far quickly becomes intractable. For example, while for simple M/M/1 or more general Markovian queuing systems the steady state is governed by the (conceptually simple) flow balance equations, transient state analysis requires the analysis of the corresponding differential equations which leads to intractable expressions even for M/M/1 systems [4]. Subsequently, either approximations or numerical methods for the transient analysis have been proposed [5] [7]. Furthermore, despite the relevance of transient analysis for communication networks, it has received little attention when analyzing networking effects. One exception example is the selection of the TCP congestion window where [8] has been considering the application of transient analysis for flows of short lengths. Through a simple, recursive formula for the average completion time of the flow transmission, the authors show a significant impact of different window settings. Nevertheless, the model does not account for queuing effects along the route, among many other aspects. A

2 second application example is the analysis of ATM networks, where [9] presented a transient analysis based on an extension of Petri nets. While demonstrating a very strong aspect of transient analysis in general, namely the ability to characterize practically relevant overload situations (which can not be dealt with by steady state analysis), the presented approach nevertheless rests heavily on numerical methods that limit the analytical insight. More contemporary approaches to network performance analysis comprise effective capacity and network calculus approaches. Effective capacity [10] was devised to provide asymptotic delay and backlog performance, initially for a Rayleigh fading channel. However, the approach is tailored towards analysis of stationary performance metrics and can provide only asymptotic results. Stochastic network calculus [11] [13] on the other hand has been employed in the literature to study the (non-asymptotic) stationary performance of wireless networks; see [14] [20]. Nevertheless, computing performance bounds for transient behaviour of wireless networks has not been attempted before. One exception is the work by Becker and Fidler [21] where they investigated certain transient aspects of networks by developing non-stationary service curves to analyze temporary phases of communication networks such as TCP slow start, sleep scheduling, and signalling in cellular networks. However, the developed nonstationary service curves were integrated into the traditional framework of stochastic network calculus. In this work, we strive to establish transient network performance analysis in particular for wireless networking. To this end, we propose the choice of state-of-the-art network calculus framework, and study different bounding techniques that allow an accurate analysis of the latency in transient settings. The major contribution of our work is to establish new performance bounds for single- and multi-hop transmission scenarios that permit the determination of upper bounds on the probabilistic end-to-end delay of a mission-critical flow in the transient regime. The rest of the paper is organized as follows; In Section II, we describe the network model and basic assumptions. In Section III, we provide some background for the problem and the used methodology. In Section IV, we formulate nonstationary performance bounds for transient network operation. In Section V, we present numerical results to evaluate the proposed bounds and compare them to simulation results for multi-hop wireless networks. We conclude in Section VI. II. NETWORK MODEL We consider a fluid-flow, discrete-time queuing model of a wireless network starting at time t 0 0. A sequence of time-critical data bits (or packets) arriving at time t 0 and lasting for T time slots traverses a network of N-hop wireless links as shown in Figure 1. The arrival is described by the cumulative arrival process A(t 0,t), 0 t 0 t where A(u,t) t 1 iu a i,t 0 u t and a i is the arrival increment during time slot i, and a i 0 for all i / [t 0,T). After being served by the system, the arrivals result in a departure process A(t) x 1 x 2 x N D(t) S 1 S 2 S N Fig. 1: Multi-hop wireless network model. D(t 0,t). We define A(t) A(t 0,t) and D(t) D(t 0,t). At time t 0, we denote the initial buffer state of the multihop network by x N +, where x n is the initial backlog of wireless link n, n 1,2,,N} and + is the set of positive integers. In this paper, we assume that A(t) obeys the (σ, ρ)-traffic characterization introduced by Cruz [22], i.e., A(t) A(u) ρ(t u)+σ, t 0 u t, (1) for some σ 0 and ρ 0. The cumulative service provided by the n th wireless link is denoted by S n (τ,t). We assume Rayleigh block fading wireless channel that provides a service process equivalent to the capacity limit, that is for channel n t 1 S n (u,t) W log 2 (1+γ n (i)). (2) iu where, W is the bandwidth and γ n (i) is the instantaneous signal-to-noise ratio (SNR) for channel n during time slot i. We assume that the services processes are i.i.d. both across links and time slots. Therefore, we write γ n (i) γy, where γ is the average SNR and the channel gain Y is an exponentially distributed random variable with unit mean. The total backlog B(t) and the end-to-end virtual delay W(t) of the queuing network described above is then given by N B(t) A(t)+ x n D(t), (3) and W(t) inf w 0 : A(t)+ } N x n D(t+w). (4) A lossless system with service process S(u, t) satisfies the input/output relationship D(0,t) A S(0,t), where is the (min, +) -convolution operator defined as X Y (u,t) inf u v t X(u,v)+Y(v,t)}. In the following, we are interested in the transient violation probability of virtual delay, given byp(w(t) > w), for a given delivery deadline w for the flow of interest (i.e. the sequence of packets) when passing through the multi-hop wireless network described above which has initial backlog x. For convenience of exposition we define τ t+w and x max max n 1,,N} x n. III. METODOLOGY AND EXISTING RESULTS The transient analysis of network performance provides a better understanding (and hence, an efficient delivery) of the quality of service requirements of incidental traffic traversing

3 multi-hop data network. This analysis is particularly important in mission-critical applications over wireless networks, such as M2M communications, wireless sensor networks and vehicular networks, that require the delivery of information of timesensitive nature and with high reliability. Hence, the problem at hand is to evaluate the transient performance (in terms of endto-end delay and reliability) of mission-critical traffic traversing a multi-hop wireless network. One possible approach for the performance analysis of such networks is based on stochastic network calculus theory [11], [12] and its recently reported application to wireless networks analysis [13]. The key benefit from using stochastic network calculus is the ability to extend single hop results to multi-hop settings with reasonable efforts. Furthermore, the described approach provides closed-form expressions in terms of the physical attributes of the wireless fading channel. Nevertheless, network calculus in general provides bounds rather than exact results, which is a necessary and acceptable compromise to achieve tractability. In the literature, there are several approaches based on stochastic network calculus to handle the performance analysis of wireless networks. These approaches range from computing the effective capacity of such channels [10] to computing the MGF of a Markov process abstraction of the wireless fading channel [20] to an ON-OFF service characterization of slotted Aloha access over shared wireless channel [19]. A more recent approach, namely (min, ) network calculus, that provides end-to-end performance characterization of wireless networks in terms of the fading channel physical parameters, i.e., fading distribution and average SNR, is developed in [13] and is based on (min, ) dioid algebra. In this paper, we pursue the transient analysis of wireless systems by utilizing the (min, ) network calculus, while we note that in principle this could also be pursued for example by MGF-based network calculus. A. Equivalent Model In order to analyze the network in Figure 1 using stochastic network calculus approach, we must first overcome the following incompatibility. In the network calculus framework it is assumed that initially all buffers are empty and no arrivals (from the considered traffic stream) has happened before the start time t 0, i.e., N x n 0 and A(0,t 0 ) 0. Furthermore, it is assumed that no service is rendered by time t 0, i.e., S(0,t 0 ) 0. To this end, we define an alternative, yet equivalent, queuing model for our system shown in Figure 2. We treat the initial backlogx n at link n as cross-traffic,a c n(t), given by A c n(t) min(κ(t t 0 ),x n ), (5) where κ(t) is a burst function with κ(t) 0, for t 0, and κ(t), otherwise. The devised model satisfies the requirements for a network-calculus-based analysis. A(t) A c 1(t) S 1 A c 2(t) S 2 A c N(t) Fig. 2: Equivalent model. For the ease of exposition we usea n (t) andd n (t) to denote the cumulative arrivals and cumulative departures, respectively, at link n. Note that A 1 (t) A(t)+A c 1(t), S N D(t) A n (t) D n 1 (t)+a c n (t), n > 1, (6) and D(t) D N (t). Given the equivalent model, without loss of generality, we use t 0 0 in the rest of the paper. B. (min, ) Network Calculus for Wireless Network Analysis The main objective of (min, ) network calculus is to obtain stationary probabilistic performance bounds for multihop wireless networks in terms of the underlying fading channel parameters. A key concept of the (min, ) network calculus is the transformation of the system model into an alternative analysis domain, known as SNR domain, using the exponential function. In this domain, the random service process rendered by a wireless fading channel is characterized in terms of the variability of the instantaneous SNR, that is the SNR service process at wireless link n is given by t 1 S n (u,t) e Sn(u,t) (1+γ n (i)) W log 2, (7) iu where we use the calligraphic font to represent processes in the SNR domain. Similarly, the cumulative arrivals and departures in the SNR domain are given by where, A n (u,t) e An(u,t) D n 1 (u,t) A c n (u,t), A c n(u,t) min(e κ(t u),e xn ), and, D n (u,t) e Dn(u,t). Then using (3), the SNR backlog process is described by B(t) e B(t) A(t) N D(t) e xn. However, the transformation does not affect time. Therefore the delay in the SNR domain is given by } N W(t) W(t) inf w 0 : A(t) e xn D(t+w). The equivalent input/output relationship in(min, )-algebra is given by D(0,t) A S(0,t), where is the (min, )- convolution operator defined as X Y(u,t) inf u v t X(u,v) Y(v,t)}. Performance analysis of communication networks often focuses on a stochastic characterization of virtual delay. As

4 shown in [13] an upper bound for the delay violation probability can be derived in terms of an integral transform, namely, the Mellin transform of the cumulative arrival and service processes in the SNR domain using the moment bound. The Mellin transform of a random process X is defined as M X (s,u,t) M X(u,t) (s) E [ X s 1 (u,t) ], (8) for any s R, whenever the expectation E[ ] exists. We restate some relevant results from [13] in the following theorem. Theorem 1 (Theorem 1, [13]). A probabilistic delay bound is given by P(W(t) > w ε ) ε, where w ε is the smallest w 0 that satisfies inf K(s,τ,t)} ε, (9) where τ t+w and K(s,τ,t) min(τ,t) u0 M A (1+s,u,t)M S (1 s,u,τ). (10) We refer to the function K(s,τ,t) above as the kernel. For a Rayleigh block fading channel operated under the assumption of the achievable Shannon capacity per slot, we compute the Mellin transform of the service increment: ] M S (1 s,i,τ) E[S s (u,τ)] E τ 1 iu τ 1 [ τ 1 iu E [(1+γ n (i)) sw log 2 iu 0 τ 1 iu 1 γ (1+γ n (i)) sw log 2 ] (1+ γy i ) sw log 2 e y i dy i ( γz) sw log 2 e z+ 1 γ dz [ e 1 γ γ ( sw log 2 Γ 1 sw log2, γ 1 )] τ u. Hence, the Mellin transform for the service process provided by a Rayleigh block fading channel is given by M S (1 s,u,τ) [V(s)] τ u, where the function V(s) is defined as follows: V(s) e 1 γ γ ( sw log 2 Γ 1 sw ) log2, γ 1. For a multi-hop wireless network with Rayleigh fading channels and (σ(s), ρ(s))-bounded cross traffic, a stationary probabilistic bound on the end-to-end delay is obtained using (min, ) network calculus [13]. C. Stationary Bound In the context of transient analysis, one may consider using the existing stationary bound provided in [13], by applying it to the equivalent model in Figure 2. However, a stationary bound cannot be directly used for our network model, because the arrivalsa i } are zero for i > T. In other words, the arrival process we consider is non-zero over a finite time horizon and Stationary bound, 5 db Stationary bound, 10 db Simulation, 5 db Simulation, 10 db End-to-end delay (ms) Fig. 3: vs end-to-end delay for a burst arrival (T 1) for a single link. x 1 0, ρ 20, σ 0. a stationary bound only makes sense for an arrival process that has non-zero arrivals over infinite time horizon. Therefore, in this work we propose to use the stationary bound as a reference by simply assuming the deterministic arrivals a i } occur over the infinite time horizon and set the corresponding parameters in the bound. Next, we describe this bound and evaluate its performance for transient network analysis. In order to obtain a bound for the stationary virtual delay, one essentially has to determine the limit of the kernel, given in (10), as t goes to infinity. Doing this, and using the introduced identities for the arrival, cross traffic and wireless service processes, we obtain: Theorem 2 (Section V-C, [13]). A probabilistic end-to-end delay bound for a cascade of N i.i.d. Rayleigh block fading channels with an average SNR of γ and(σ c,ρ c )-bounded cross traffic is given by e s( ρw+σ+nσ c) P(W(t) > w) inf (1 V 0 (s)) s. } min1,(v 0 (s)) w (w+1) }, where V 0 (s) e s(ρ+ρc) V(s). In order to apply this result to the model described above, we dimension the cross traffic at each link using Eq. (5) by setting σ c x max and ρ c 0. We refer to the upper bound given in Theorem 2 as stationary bound in the rest of the paper. To evaluate the viability of this approach, we compare, in Figure 3, the transient violation probability obtained using simulation with the stationary bound for a single link. We use 1 ms slot duration and simulate the system with a burst arrival of size 20 bits and no initial backlog, i.e., x 0. We make a note that, considering a burst only process for computing the stationary bound does not yield any meaningful bound. Therefore, as mentioned before we set the parameter ρ 20

5 by assuming a constant arrival process with rate 20 bits per time. Taking different target latencies into account, as well as different settings of the average SNR of the wireless link, from Figure 3 we observe that the stationary bound is arbitrarily loose. Further, the bound deviates from the simulation for higher end-to-end delay. It thus becomes immediately clear that the stationary bound is by no means a good analytical expression for instance to choose from different routes or to steer resource allocation to optimize the transient violation probability for critical machine-type data, as an arbitrary slackness is introduced. We thus strive for better bounds, as presented in the following. IV. TRANSIENT ANALYSIS In this section, we present our main contribution, namely, new transient upper bounds for the violation probability. To this end, we present two different approaches and later (in Section V) compare the probabilistic delay bounds resulting from both approaches. A first, straightforward approach is to start from the definition of the kernel as presented in Section III-B and derive a transient bound. We refer to this as State Of The Art (SOTA) transient bound, as it is essentially an application of known results. However, as we will show later, this bound - while improving over the stationary bound - is still loose. This motivates us to present new analysis method for the transient behaviour of the system which resulted in the new proposed transient bound. This proposed transient bound is our main contribution. A. State Of The Art (SOTA) Transient Bound In this subsection, we derive a transient upper bound for the violation probability using the results from [13]. We first note that the bound on the violation probability in Theorem 1 is indeed applicable also to transient analysis. We hence focus on the components of the kernel. Since the arrival process is deterministic and is (σ, ρ)-bounded (1), we have 1 s 1 logm A(s,u,t) ρ (t u)+σ M A (1+s,u,t) e s(σ+ρ(t u)). We note that when u t we have M A (1+s,u,t) 1. We incorporate this fact as follows: M A (1+s,u,t) e s(σmin1,t u}+ρ(t u)). (11) LetS 0 denote the dynamic SNR server that describes the net service offered by the multi-hop route to the through traffic. In the following lemma we present the Mellin transform of S 0. Lemma 1 (Lemma 6, [13]). Given σ c x max and ρ c 0, the Mellin transform of S 0 (u,τ) satisfies for s < 1 that ( ) N 1+τ u M S0 (1 s,u,τ) e snxmax (V 0 (s)e sρ ) τ u. τ u (12) Substituting (11) and (12) into the definition of the kernel, and with some manipulation we obtain: K(s,τ,t) e s(nxmax ρw)[ t 1 ( ) N 1+τ u e sσ V0 τ u (s) τ u u0 ( ) N 1+w + [V 0 (s)] w]. (13) w Using the results above, we can state the SOTA transient delay bound in the following corollary. Corollary 1. For a single wireless link, an upper bound for P(W(t) > w) is given by min e s(x1 ρw) (V 0 (s)) w[ e sσ V0(s) (V 0 (s)) t+1 +1] }. 1 V 0 (s) Proof. Using Theorem 1 and substituting N 1 in (13) we obtain P(W(t) > w) min e s(xmax ρw)[ t 1 e sσ V 0 τ u (s)+(v 0 (s)) w]} u0 min e s(x1 ρw)[ e sσ V0 τ (s) ( 1 V 0(s) )t 1 ] } 1 ( V ) 1 +(V 0(s)) w 0(s) min e s(x1 ρw) (V 0 (s)) w[ e sσv 0(s) (V 0 (s)) t+1 +1] }. 1 V 0 (s) Later, in Section V we will show that for multi-hop scenario with non-zero backlogs the SOTA transient bound becomes very loose. The SOTA transient bound slackness is mainly due to the fact that it is based on results that are initially derived for stationary settings. B. The Proposed Transient Bound Our proposed transient bound is inspired by the bounding techniques used in [13]. However, we conduct independent analysis starting with the basic principles of network calculus and tailor the result, from the beginning, to our system with initial backlog. We note that our analysis is more involved due to the presence of the initial backlog x. We start by presenting our analysis for a single-hop scenario, i.e., for N 1. Then we generalize the obtained results for the multi-hop, N > 1, case. In the following theorem, we state the proposed upper bound for the single hop case. Theorem 3. For a single-hop scenario, an upper bound for PW(t) > w} is given by [ ] t 1 A s (t)e sx V τ (s)+ [A(t)/A(u)] s V τ u (s). min Proof. From network calculus, we have D(τ) min 0 u τ [S(τ u) A(u) Ac 1 (u)]. Recall that A c 1(0) 1 and A c 1(u) e x1 for all u > 0. Now, the event W(t) > w} is equivalent to the event

6 that the cumulative departures at time τ is strictly less than the cumulative arrivals at time t plus the initial backlog x 1. Therefore, PW(t) > w} PD(τ) < A(t)e x1 } P min [S(τ 0 u τ u) A(u) Ac 1 (u)] < } A(t)ex1 τ PS(τ) < A(t)e x1 } ( S(τ u) A(u) < A(t)})} t 1 PS(τ) < A(t)e x1 } ( S(τ u) A(u) < A(t)})} t 1 PS(τ) < A(t)e x1 }+ PS(τ u) A(u) < A(t)} [ min A s (t)e sx1 E[S s (τ)]+ [ min t 1 [ ] s A(t) E[S s (τ u)]] A(u) t 1 [ ] s A(t) A s (t)e sx V τ (s)+ V (s)] τ u. (14) A(u) In the fourth step above we have used the fact thatps(τ u) A(u) < A(t)} 0 for u t. In the fifth step we have used the union bound. In the sixth step we have used the moment bound. Next, we present transient bound for the N-hop case. Even though the analysis for the N-hop case is more involved, it essentially uses the same bounding techniques from the proof of Theorem 3. Theorem 4. For the N-hop network in Figure 2, we compute PW(t) > w} Φ, where Φ is given by (15) Proof. The proof is given in the Appendix. V. NUMERICAL ANALYSIS In this section we first present a comparison of the proposed transient bound with stationary bound and the SOTA transient bound. We then evaluate the tightness of the proposed transient bound in comparison with the violation probability obtained using simulation. We use 1 ms slot duration and the default parameter values are as follows: ρ 25,σ 25, SNR 5 db, and equal backlogs at each node with total initial backlog equal to 100. We consider two arrival processes, 1) a burst arrival with T 1,σ 25,ρ 0, which may model a control packet that entered the network at t 0, and 2) a constant arrival process with T 5,σ 0,ρ 25, which may model the data that arrives in five time slots. The numerical bounds are computed using MATLAB and the Discrete Event Simulation is done using C. A. Comparison of Upper Bounds Recall that the stationary bound cannot be directly applied to the problem at hand. We use it as a reference by assuming the arrival process occurs over infinite time horizon and accordingly set the corresponding parameters in the bound. From Figure 4, we observe that both the proposed transient and SOTA-Transient are significantly lower than the stationary bound. Note that the proposed transient is not significantly lower than the SOTA transient for this simplest case of burst arrival. However, the difference increases by half an order of magnitude for constant arrival process with T 5 as shown in Figure 5. In Figures 6 and 7, we further compare the bounds for a two-hop network for SNR values 5 db and 10 db, respectively. We note that for the two-hop network the SOTA transient bound performs worse than that for a one hop case. Finally, in Figure 7, for average SNR 10 db, i.e., at lower utilization (43%), the proposed transient bound is tighter by two orders of magnitude compared with the SOTA transient bound. The results above demonstrate that the proposed transient bound significantly outperforms the SOTA transient and stationary bound under more general settings involving multiple hops, large initial backlogs and different utilizations. Therefore, we conclude that the SOTA transient bound that is derived directly using the results from stationary analysis is inadequate for transient analysis. B. Evaluation of the Proposed Transient In the previous section, we have seen that the proposed transient bound is within one order of magnitude of the simulated violation probability. In this section, we further investigate its performance for different parameter settings. In Figures 8 and 9, we present performance results by varying the average SNR and the total initial backlog, respectively, in a two-hop network. It is interesting to note that the gap between the simulated bound and the proposed transient bound is around half an order of magnitude for a wide range of parameters. In Figure 10, we compare the end-to-end delay in a threehop network. In this case, the gap increases beyond one order of magnitude. We expect this trend to continue as the number of hops increase. However, from all the figures above, we infer that the proposed transient bound has a decay rate that always matches closely with the decay rate of the simulated violation probability. The significance of this property is that, an optimization of the proposed transient bound can yield good heuristic solutions for the optimization of the end-to-end delay in the network operating in transient state. VI. CONCLUSIONS We have studied the problem of characterizing the endto-end delay of a sequence of time-critical control packets traversing through a multi-hop wireless network with non-zero initial backlog at each hop. As this requires the network to be analysed in the transient state, we attempt to find upper bounds for the end-to-end delay using stochastic network calculus. We have studied the state-of-the-art upper bounds and have demonstrated their poor performance for the problem at hand. We have derived new transient bounds by using the first principles of network calculus and the state-of-the-art bounding techniques. Through extensive simulations we have showed that the proposed transient bound is significantly better

7 τ Φ min V τ (s)[a(t)] s u u 11 min(u N 2,t 1) u 1 [A(u )] s V u (s)+ i0 ( i+τ 1 τ 1 ) e s N i xn. (15) Stationary bound, x1 100 Stationary bound, x1 0 SOTA transient, x1 100 SOTA transient, x1 0 Proposed transient, x1 100 Proposed transient, x1 0 Simulation, x1 100 Simulation, x1 0 Single-hop end-to-end delay (ms). SNR 5 db Fig. 4: versus end-to-end delay for a burst arrival (T 1) and different backlogs. ρ 0 and σ 25. Stationary bound SOTA transient Proposed transient Simulation Single-hop end-to-end delay (ms). SNR 5 db Fig. 5: versus end-to-end delay for an arrival process with T 5. x 1 100,ρ 25 and σ 0. than the alternatives. Also, we have observed that its decay rate closely matches the decay rate of the simulated violation probability. REFERENCES [1] O. N. C. Yilmaz, Y. P. E. Wang, N. A. Johansson, N. Brahmi, S. A. Ashraf, and J. Sachs, Analysis of ultra-reliable and low-latency 5g communication for a factory automation use case, in 2015 IEEE International Conference on Communication Workshop (ICCW), June 2015, pp [2] E. Dahlman, G. Mildh, S. Parkvall, J. Peisa, J. Sachs, Y. Seln, and J. Skld, 5g wireless access: requirements and realization, IEEE Communications Magazine, vol. 52, no. 12, pp , December [3] J. Hedberg, Safety requirements specifications?guideline, SP Swedish National Testing and Research Institute, Stationary bound SOTA transient Proposed transient Simulation Two-hop end-to-end delay (ms), SNR 5 db Fig. 6: versus end-to-end delay for an arrival process with T 5. x n 100,ρ 25 and σ 0. Stationary bound SOTA transient Proposed transient Simulation Two-hop end-to-end delay (ms). SNR 10 db Fig. 7: versus end-to-end delay for an arrival process with T 5. x n 100,ρ 25 and σ 0. [4] P. Morse, Queues, Inventories and Maintenance: The Analysis of Operational Systems with Variable Demand and Supply. Wiley, [5] J. Zhang and E. J. Coyle, The transient solution of time-dependent m/m/1 queues, IEEE Transactions on Information Theory, vol. 37, no. 6, pp , Nov [6] T. Matis and R. Feldman, Transient Analysis of State-Dependent Queuing Networks via Cumulative Functions, Journal of Applied Probability, vol. 38, [7] T. Czachrski, Queueing Models for Performance Evaluation of Computer Networks - Transient State Analysis. Springer International Publishing Switzerland, [8] M. Mellia and H. Zhang, Tcp model for short lived flows, IEEE Communications Letters, vol. 6, no. 2, pp , Feb [9] C. Wang, D. Logothetis, K. Trivedi, and I. Viniotis, Transient Behavior of ATM Networks under Overloads, in IEEE Infocom, March [10] W. Dapeng and R. Negi, Effective Capacity: A Wireless Link Model

8 Proposed transient, w 3 Proposed transient, w 6 Simulation, w 3 Simulation, w 6 Proposed transient, T 1 Proposed transient, T 5 Simulation, T 1 Simulation, T SNR (db) Fig. 8: versus average SNR for different w. N 2,x n 50,ρ 25 and σ 0. Three-hop end-to-end delay (ms). SNR 5 db Fig. 10: versus end-to-end delay for different T. N 3,x n 33,ρ 25 and σ Proposed transient, w 6 Proposed transient, w 9 Simulation, w 6 Simulation, w Total initial backlog Fig. 9: versus total initial backlog for different w. N 2,ρ 25 and σ 0. for Support of Quality of Service, IEEE Transactions Wireless Communications, vol. 2, no. 4, pp , July [11] Y. Jiang, A Basic Stochastic Network Calculus, SIGCOMM Comput. Commun. Rev., vol. 36, no. 4, pp , Aug [12] M. Fidler, An end-to-end probabilistic network calculus with moment generating functions, in Quality of Service, IWQoS th IEEE International Workshop on. New Haven, CT: IEEE, 2006, pp [13] H. Al-Zubaidy, J. Liebeherr, and A. Burchard, Network-layer performance analysis of multihop fading channels, IEEE/ACM Transactions on Networking, vol. 24, no. 1, pp , Feb [14], A (min, ) Network Calculus for Multi-Hop Fading Channels, in IEEE Infocom. Turin: IEEE, 2013, pp [15] N. Petreska, H. Al-Zubaidy, and J. Gross, Power Minimization for Industrial Wireless Networks under Statistical Delay Constraints, in Teletraffic Congress (ITC), th International. Karlskrona: IEEE, Sept 2014, pp [16] N. Petreska, H. Al-Zubaidy, R. Knorr, and J. Gross, On the Recursive Nature of End-to-End Delay Bound for Heterogeneous Wireless Networks, in IEEE International Conference on Communications 2015 (ICC 2015). London: IEEE, June 2015, pp [17] K. Zheng, F. Liu, L. Lei, C. Lin, and Y. Jiang, Stochastic Performance Analysis of a Wireless Finite-State Markov Channel, IEEE Transactions Wireless Communications, vol. 12, no. 2, pp , February [18] F. Ciucu, R. Khalili, Y. Jiang, L. Yang, and Y. Cui, Towards a System Theoretic Approach to Wireless Network Capacity in Finite Time and Space, in IEEE Infocom. Toronto: IEEE, April 2014, pp [19] F. Ciucu, Non-Asymptotic Capacity and Delay Analysis of Mobile Wireless Networks, SIGMETRICS Perform. Eval. Rev., vol. 39, no. 1, pp , Jun [20] M. Fidler, WLC15-2: A Network Calculus Approach to Probabilistic Quality of Service Analysis of Fading Channels, in Globecom 06. IEEE. San Francisco: IEEE, Nov 2006, pp [21] N. Becker and M. Fidler, A non-stationary service curve model for performance analysis of transient phases, in th International Teletraffic Congress, Sept 2015, pp [22] R. L. Cruz, A calculus for network delay. i. network elements in isolation, IEEE Transactions on Information Theory, vol. 37, no. 1, pp , Jan A. Proof of Theorem 4 APPENDIX Again, using network calculus we have PW(t) > w} PD N (τ) < A(t) e N xn } PA N S N (τ) < A(t) e N xn } P(D A c N ) S N(τ) < A(t) e N xn } P min [D (u) A c N (u) S N(τ u)] < A(t) e N 0 u τ P S N (τ) < A(t) e N xn } τ ( D (u) S N (τ u) < A(t) e } xn }) xn } PS N (τ) < A(t) e N xn } τ + PD (u) S N (τ u) < A(t) e xn }}. (16) In the following we find an upper bound for the probabilities in the summation term of the RHS of (16). Noting that

9 PW(t) > w} τ u i0 u 11 + τ u u 11 u i 2 u i 11 u N 2 u 1 i 1 P S N i (u i 1 ) S N n (u n 1 u n ) S N (τ u) < A(t) e } N i xn } P A(u ) S N n (u n 1 u n ) S N (τ u) < A(t). (18) τ u i0 u 11 τ u i0 u 11 min u i 2 u i 11 u i 2 u i 11 i 1 PS N i (u i 1 ) [A(t)] s e s N i xn V τ (s) i0 S N n (u n 1 u n ) S N (τ u) < A(t) e N i xn } min [A(t)]s e s( N i i 1 xn) E[S N i (u i 1 ) S N n (u n 1 u n ) S N (τ u)} s ] τ u u 11 u i 2 u i 11 1 min i0 ( i+τ 1 τ 1 ) [A(t)] s e s N i xn V τ (s). (19) τ u u 11 τ u u 11 τ u u 11 min τ u N 2 u 1 PA(u ) S N n (u n 1 u n ) S N (τ u) < A(t)} min(u N 2,t 1) u 1 min(u N 2,t 1) u u 11 u 1 min(u N 2,t 1) u 1 PA(u ) min [A(t)/A(u )] s E[ S N n (u n 1 u n ) S N (τ u) < A(t)} S N n (u n 1 u n ) S N (τ u))} s ] [A(t)/A(u )] s V τ u (s). (20) D (u) A S (u), we have PD (u) S N (τ u) < A(t) e xn } P(D N 2 A c ) S (u) S N (τ u) < A(t) e xn } PS (u) S N (τ u) < A(t) e xn }+ u PD N 2 (u 1 ) S (u u 1 ) S N (τ u) < A(t) e N 2 xn }. u 1 1 Substituting (17) in (16), we obtain PW(t) > w} PS N (τ) < A(t) e N xn }+ (17) PS (u) S N (τ u) < A(t) e xn }+ τ u P D N 2 (u 1 ) S (u u 1 ) S N (τ u)<a(t) e } N 2 xn. u 1 1 One can again use similar manipulation as in (17) to bound the probabilities in the double summation of the RHS of the above inequality. Now, repeating the above step iteratively, we arrive at (18). The first and second terms in the RHS of (18) are upper bounded as shown in (19) and (20), respectively. Finally, substituting (19) and (20) in (18), the theorem follows.