A Network Calculus Approach to Probabilistic Quality of Service Analysis of Fading Channels

Transcription

1 A Network Calculus Approach to Probabilistic Quality of Service Analysis of Fading Channels Markus Fidler Department of Electrical and Computer Engineering University of Toronto, Ontario, Canada Abstract Network calculus is an established theory for deterministic quality of service analysis of fixed networks. Due to the failures inherent in fading channels it is, however, not applicable to radio systems. Emerging probabilistic equivalents allow closing this gap. Based on the recent network calculus with moment generating functions we present a methodology for performance analysis of fading channels. We use a service curve representation of radio links which facilitates an efficient analysis of radio networks. We investigate fading channels with memory and our results show that the fading speed impacts service guarantees significantly. Numerical performance bounds are provided for an example taken from cellular radio communications for which the effects of opportunistic scheduling are quantified. Simulation results are shown which confirm the efficiency of the approach. I. INTRODUCTION Network calculus [4], [18] is a theory of deterministic queuing systems that is based on the early (σ, ρ)-calculus for network delay presented in [8] and on the work on generalized processor sharing in [1]. It facilitates the efficient derivation of backlog and delay bounds applying upper envelopes on traffic arrivals and lower bounds on the offered service, socalled arrival and service curves. The deterministic view of network calculus generally considers the worst-case. Consequently, respective performance bounds cannot be provided for radio systems, where due to the failures inherent in fading channels deterministic service guarantees become meaningless. Here, performance bounds have to be complemented with certain violation probabilities to incorporate nondeterministic service provisioning. Considerable effort has been made in recent time to develop a probabilistic equivalent of network calculus. A probabilistic (σ(θ),ρ(θ))-calculus was derived in [4], [3]. Related models were used in [8], [5], [9] where the concepts of exponentially, stochastically, and generalized stochastically bounded burstiness were introduced respectively. In [1] a theory using effective envelopes was developed. Related models using second order statistics were proposed in [17], []. Effective envelopes were further investigated in [], [0], [7], and in [19] where the relation to the theory of effective bandwidths, for example [4], [14] was shown. The theory of effective bandwidths provides a rich variety of probabilistic traffic models which are based on moment generating functions (MGFs). A probabilistic network calculus with MGFs was proposed in [4] and further elaborated in [11]. This work was supported by the DFG under an Emmy Noether grant. Few attempts have, however, been made to apply probabilistic network calculus to derive statistical service guarantees for radio communication systems. Related models are crucial for efficient management of scarce radio resources with respect to quality of service requirements. A well-established basis for such higher-layer models is the Gilbert-Elliott channel [1], [9] which describes time-varying channel conditions using Markov chains. Further details are given for example in [6]. Of particular interest in the context of this work are models for radio block error processes as elaborated in [30]. The Gilbert-Elliott channel with two states, good and bad, was applied in [16] and combined with an on-off fluid source. The resulting four state Markov model is decomposable and can thus be solved, however, the approach is limited to Markov traffic sources. In [7] the concept of wireless effective capacity, which is related to the theory of effective bandwidths, was developed and further elaborated in [4]. The work in [4] builds on a memoryless on-off channel with constant rate traffic sources and introduces models for opportunistic scheduling. In contrast, network calculus provides a general decoupling of arrival and service processes which allows combining a wide variety of traffic and service models beyond the mentioned special cases. To our knowledge, the first application of wireless link models in the context of network calculus is presented in [13] using an on-off impairment model. In this work we derive a service curve model for Gilbert- Elliott channels with memory. We provide explicit results on the impact of the fading speed on service guarantees, where we show that fast-fading channels allow providing significantly better performance bounds than slow-fading channels. Further on, we show that the quality of service problem of slowfading channels can hardly be resolved by allocation of extra resources. These effects are of significant importance and relate to the performance of opportunistic scheduling. The remainder of this paper is organized as follows: In Sect. II we provide the required background on probabilistic network calculus. Sect. III introduces models for radio communication systems and Sect. IV provides a numerical evaluation for the General Packet Radio Service (GPRS). II. PROBABILISTIC NETWORK CALCULUS This section provides a brief overview on probabilistic network calculus with MGFs [4], [11]. We use a discrete time model with t N 0 = {0, 1,,...}. Arrival and departure

2 processes are described by real-valued cumulative functions A(0,t) and D(0,t) respectively, which represent the amount of data observed in the interval (0,t]. Clearly, A(0,t) and D(0,t) are nonnegative and increasing in t. The amount of data seen in an interval (s, t] is denoted by real-valued bivariate functions A(s, t) =A(0,t) A(0,s) and D(s, t) = D(0,t) D(0,s) respectively, where A(s, t) and D(s, t) are nonnegative, increasing in t, and decreasing in s. Let A(0,t) and D(0,t) be the arrival respectively departure process of a system. Assume S(s, t) is a random process that denotes the service offered by the system in the interval (s, t] where S(s, t) is nonnegative, increasing in t, and S(t, t) =0 for all t 0. The system is called a dynamic server if for all t 0 it holds that D(0,t) inf [A(0,τ)+S(τ,t)]. (1) τ [0,t] The definition of dynamic servers stems from [4], [5], [6]. It extends the framework of deterministic network calculus using bivariate functions. Performance bounds for dynamic servers have been obtained in [4], [5], [6] where it has also been shown that dynamic servers in series can be effectively collapsed into an equivalent single server. Thus, known performance bounds extend immediately to tandem servers. An important class of systems that fulfill (1) are work-conserving servers with a time-varying capacity [5]. In [4], [11] a network calculus with MGFs is developed using the concept of dynamic servers. Throughout this work we assume that arrivals and service are described by statistically independent, stationary random processes. Under stationarity A(s, s + t) equals A(0,t) in distribution for all s 0. The MGF of a stationary random process A is defined as M A (θ, t) =Ee θa(0,t). We use the notation M S (θ, t) = M S ( θ, t). MGFshavea number of useful properties. For additive and multiplicative constants a and b it is known for all θ that M a+ba (θ, t) =e θa M A (bθ, t) () and for the addition of two statistically independent random processes A and B it holds for all θ that M A+B (θ, t) =M A (θ, t)m B (θ, t). (3) The following important results, which were first proposed in [4], establish a probabilistic network calculus with MGFs. Respective proofs can be found in [11]. Consider two dynamic servers S 1 (s, t) and S (s, t) in series. There exists an equivalent single dynamic server S(s, t) = inf τ [s,t] [S 1 (s, τ)+s (τ,t)]. Assume S 1 (s, t) and S (s, t) are statistically independent, stationary, and have MGF M S1 (θ, t) respectively M S (θ, t). The MGF of the equivalent single server is upper bounded for t 0 and all θ according to M S (θ, t) t M S1 (θ, τ)m S (θ, t τ). τ=0 Note that this step can be applied iteratively to derive the MGFs that correspond to the end-to-end concatenation of an arbitrary number of dynamic servers in series. Performance bounds [4], [11] follow with Chernoff s theorem. Consider a dynamic server S(s, t) with arrival process A(s, t). Assume S(s, t) and A(s, t) are statistically independent, stationary, and have MGF M S (θ, t) respectively M A (θ, t). An upper backlog bound and assuming first-come first-served ordering an upper delay bound that are violated at most with probability ε (0, 1] are )] b =inf θ>0 [ 1 θ ( ln [ [ d =inf inf τ : 1 ( ln θ>0 θ s=0 M A (θ, s)m S (θ, s) ln ε ) M A (θ, s τ)m S (θ, s) ln ε s=τ ]] 0. The infinite sums can be solved using geometric series if bounds on the MGF of the arrival and service process, for example of the form M A (t) e θ(σ(θ)+ρ(θ)t) [4] and M S (t) e θρs( θ)[t T ]+, are known, where ρ(θ) <ρ S ( θ) for some θ>0 is required for stability [4], [11]. III. FADING CHANNEL MODELS We provided the required background on probabilistic network calculus with MGFs in Sect. II. The presented framework is of particular interest because MGFs for a considerable variety of different traffic models are known from the theory of effective bandwidths, see for example [4], [14], including periodic sources, on-off fluid sources [10], [15], and extremal shape controlled traffic [1]. Due to the failures inherent in fading channels radio links have not been easily accessible to network calculus and the derivation of quality of service guarantees for radio systems has proven to be challenging. Essential for the application of such methods are probabilistic service curves that describe fading channels. Respective models are not well-investigated, however, in the sequel we show that under the concept of dynamic servers corresponding MGFs of radio channel models follow intuitively. An important feature of abstracting higher-layer models of fading channels is the consideration of memory between consecutive transmissions which can be reproduced by Markov chains. Respective models were first considered in [1], [9] and named Gilbert-Elliott model after the originators. Fig. 1 shows a two-state, simplified Gilbert-Elliott model using discrete time, where the channel can either be in on state, that is data are decoded error-free, or in off state, in which case data cannot be decoded correctly at the receiver. A number of further promising studies of related models have been conducted. The state space has been extended in [6] to generate more accurate results for flat Rayleigh fading channels under drastic variations. In [30] the block error process is investigated for flat Rayleigh or Rician fading channels to complement former studies of the symbol error process and a two-state Markov chain has been found to be sufficiently accurate.

3 Fig p off p q on 1-q Discrete time, two-state, simplified Gilbert-Elliott model. In a related context Markov models have been used extensively to describe arrival processes, for example of voice audio sources, and corresponding MGFs have been derived in [10], [15], [4]. We reuse these models to describe random processes S(s, t) that denote the service offered by a fading channel in the interval (s, t]. Under the assumption that the channel is a work-conserving server with a time-varying capacity it can be seen as a dynamic server according to (1). Consider an irreducible, homogeneous Markov chain with n states and stationary state distribution π, where π is a row vector. Workload is processed with rate h i when in state i. Under a discrete-time model let H(θ) be the diagonal matrix diag(e θh1,e θh,...,e θhn ),letq be the transition probability matrix, where q ij is the transition probability from state i to state j, and 1 is a column vector of ones. For all t 0 and all θ the MGF of the corresponding random process S is [4] M S (θ, t) =π(h( θ)q) t 1 H( θ)1. (4) Although the Markov block error model uses discrete time it may in some cases be approximated with reasonable precision by the continuous-time model. In this case let H = diag(h 1,h,...,h n ) and let Q be the transition rate matrix. For all t 0 and all θ the MGF of the corresponding random process S is [10], [15], [14] M S (θ, t) =πe (Q θh)t 1. (5) Closed-form solutions of (4) and (5) for models with only two states exist for t [14], [4]. For two-state, continuoustime models a closed-form bound has been reported for all t 0 in [7]. Consider the continuous-time model in (5) with two states, on and off. Denote the elements of Q by q ij where q 11 = µ, q 1 = µ, q 1 = λ, q = λ and λ, µ > 0. It follows that π = (λ/(λ + µ),µ/(λ + µ)). Workload is processed only in the on state (state 1) with rate h. For all t 0 and all θ it holds that M S (θ, t) e θρs( θ)t where ρ S (θ) = 1 ( (λ µ + hθ) +4λµ λ µ + hθ). (6) θ The exact solution is provided in (7) in the appendix. Given a two-state, continuous-time Markov channel with M S (θ, t) e θρs( θ)t and a (σ(θ),ρ(θ))-constrained arrival process with M A (θ, t) e θ(σ(θ)+ρ(θ)t) closed form backlog and delay bounds can be derived using the approach in [4], [11]. Under first-come first-served scheduling the delay bound is the smallest τ that fulfills ( 1 ) ln e θ(σ(θ)+ρ(θ)(s τ) ρs( θ)s) ln ε 0 θ s=τ which can be solved for τ if ρ(θ) < ρ S ( θ) using the geometric series. It follows that τ σ(θ) ln γ ln ε + ρ S ( θ) θρ S ( θ), where γ = 1 1 e θ(ρs( θ) ρ(θ)) and d =inf θ>0 [inf[τ]] is the resulting delay bound. The backlog bound follows along the same line as b =inf θ>0 [σ(θ)+ (ln γ ln ε)/θ]. a) Impairment Model: A related channel model is also investigated in [13], where a constant rate server is readjusted to match the two-state channel with on and off state by introducing the concept of an impairment process. While in on state the impairment process consumes the entire available service of the server and while in off state it does not consume any service. The MGF of the impairment process is bounded according to M A (θ, t) e θρ(θ)t where ρ(θ) is given by (6), however, we exchange λ and µ such µ is the rate of transitions into the on state and λ into off state, such that the on state of the impairment process corresponds to the off state of the resulting channel model. The MGF of the constant rate server is M S (θ, t) =e θht and the MGF of the remaining service M S A (θ, t) after the impairment can be derived with () and (3) as M S A (θ, t) =M S (θ, t)m A (θ, t), such that ρ S (θ) c = 1 θ ( ( λ+µ+hθ) +4λµ λ µ+hθ) M S A (θ, t) e θht+ t ( = e t (λ µ hθ) +4λµ λ µ hθ). Thus, the impairment model yields the same bound on the MGF of the service as the two-state channel model in (6). b) Fading Speed: Of particular relevance for the characterization of the Gilbert-Elliott model is the speed of the fading process. Highly correlated processes are referred to as slowly fading in contrast to fast fading which refers to processes with small memory. Consider a continuous-time, two-state Markov channel according to (6). We investigate the case where λ and µ are scaled by a constant c which determines the fading speed while maintaining a non-varying steady state error probability cµ/(cλ + cµ) = µ/(λ + µ). Substituting λ and µ by cλ and cµ respectively the first derivative of (6) becomes ( (cλ cµ + hθ)(λ µ)+4cλµ ). (cλ cµ + hθ) +4c λµ λ µ After some conversions it can be seen that θ ρ S (θ)/ c is negative if λ, µ > 0 and h 0. Thus ρ S (θ) is decreasing in c if θ > 0 and increasing in c if θ < 0. Consequently, performance bounds improve if the fading speed increases. c) Opportunistic Scheduling: If different terminals see channels with independent fading processes an opportunistic scheduler, which generally assigns the channel to the terminal seeing the best, instantaneous channel conditions, can maximize the throughput. Related models are also investigated in [4], though using a memoryless on-off channel. In a practical implementation the channel condition in the near future will be estimated from past measurement reports. The accuracy of this prediction becomes better if the memory of the channel increases.

4 TABLE I PARAMETERS USED THROUGHOUT SECT. IV. Parameter Value Description q/(p + q) 0.1 steady state block error probability p 0.1 transition probability from off to on state h 4 peak channel rate τ 8 period of the source σ 8 burst size of the source ε 10 3 delay bound violation probability We use a model for opportunistic scheduling for two terminals with independent fading channels, which each are described by a two-state Gilbert-Elliott model. The combined four-state model has states (off,off), (on,off), (off,on), and (on,on). We assume that resources are always allocated to the terminal which sees a good channel and in case both terminals see a good channel the channel rate h is evenly divided among the two terminals. If terminal is greedy than the service offered to terminal 1 can be described by (4) where H(θ) =diag(1,e θh, 1,e θh/ ). Models for more than two terminals follow accordingly by increasing the state space. IV. NUMERICAL RESULTS We provide numerical results for a channel described by the discrete-time, two-state Markov model in Fig. 1 and an independent, periodic traffic source. The source generates σ units of workload at times {Uτ + nτ, n =0, 1,...} where τ is the period of the source and U is the initial start time which is uniformly distributed in the interval [0, 1]. For all t 0 and θ 0 it is known, see [14] and references therein, that M A (θ, t) =e θσ ( ( ) ) t τ t t 1+ τ (e θσ 1). τ The channel model is specified by the transition rate from off to on state p and the steady state block error probability q/(p+q) which for given p determines the transition rate from on to off state q. Initially we choose p =0.1 and q/(p + q) = 0.1 to model a slowly fading channel. For an evaluation and discussion of characteristic values see [30]. Tab. I summarizes the parameters used unless stated otherwise. Delay bounds are computed using the probabilistic network calculus shown in Sect. II, where we optimize the parameter θ numerically. The infinite sum in the delay bound formula is computed for the first 000 units of time and the tail is estimated using the geometric series. In the context of GPRS the numerical values can be interpreted as following: A Radio Link Control (RLC) block consists of four bursts which are transmitted in consecutive Time Division Multiple Access (TDMA) frames. Each TDMA frame is of ms duration and consists of eight bursts which, however, belong to different RLC blocks. Taking the GPRS multi-frame structure into account, one RLC block is transmitted approximately each 0 ms, which is chosen as the base unit of the discrete time model here. Consequently, relative error violation probability ε Fig.. simulation analysis relative error Comparison of analytical bounds and simulation results. the rate h =4used for the channel model corresponds to a GPRS multi-slot class where four time-slots can be used in parallel, such that four RLC blocks can be transmitted virtually in parallel within 0 ms. Unused time-slots can be used to schedule blocks with lower quality of service requirements. Note that delay bounds are derived assuming first-come first-served ordering. Possible effects due to automatic repeat request at the RLC layer and reordering of RLC blocks are not taken into account. Also, delays which are due to Medium Access Control (MAC) and related signalling are not considered here. Such delays may be negligible in case of the GPRS downlink, however, in case of uplink transmission there may be an additional impact mainly due to random access and contention resolution. The traffic pattern generated by the periodic source model relates to the structure of video traffic. The source that is used here can be interpreted as a video source that generates a frame which consists of 8 RLC data blocks every 8 units of time that is every 160 ms. Assuming an encoding using the GPRS Coding Scheme 3 (CS-3) radio link blocks have a payload of approximately 36 B, video frames are of 88 B size, and the source data rate becomes 14.4 kb/s. At first we investigate the accuracy of probabilistic delay bounds which are derived based on [4], [11] using Chernoff s inequality. Fig. shows numerical delay bounds for violation probabilities ε [ ] which are compared to the distribution function obtained by simulation of units of time respective 10 9 source periods. The simulation model builds on the same Markov channel as the analytical model to quantify the accuracy. The relative error d num /d sim 1 is included in Fig., which confirms that the numerical bounds become reasonably tight, if the violation probability is small. The memory of the fading process with respect to the radio block length, which is reflected by the transition probabilities of the Markov block error model, has a significant impact on performance bounds. The model in [30] depends on the delay bound [timeslots]

5 delay bound [timeslots] h = h = 4 h = fading speed p with q/(p+q) = 0.1 delay bound [timeslots] opportunistic scheduling processor sharing delay bounds stability region block error probability q/(p+q) with p q = 1/ rate [blocks/timeslot] Fig. 3. Impact of the fading speed respectively Doppler frequency. Fig. 4. Opportunistic scheduling. normalized Doppler frequency f D NT, that is terminal velocity divided by carrier wavelength f D multiplied by the block distance NT, where N is the number of symbols per block and T is the symbol duration. As an example the parameters p =0.5 and q/(p + q) =0.1 can under certain conditions be approximately assumed for a normalized Doppler frequency f D NT =0.05 [30]. In the context of GPRS with NT 0 ms this corresponds to a Doppler frequency of f D =.5 Hz and a terminal speed of 0.83 m/s in case the 900 MHz band with a wavelength of 0.33 m is used. Thus, the channel has comparably small memory and a small average burst error length already for low terminal speed. The impact of the fading speed, which determines p, is shown in Fig. 3 for a constant steady state block error probability q/(p + q) = 0.1. A significant impact of the fading speed on the delay bounds can be noticed and as established in Sect. III for the continuous-time, two-state Markov channel, delay bounds improve if the fading speed increases. In addition, Fig. 3 shows the impact of the peak channel rate h, where h [, 4, 8], on delay bounds. It is obvious, that providing a higher rate improves delay bounds, however, the relative improvement depends largely on the fading speed. For slow fading the gain obtained from an allocation of extra resources is comparably small. With respect to the investigated scenario this result indicates that the duration of the off periods is the determining factor under slow fading. When increasing the rate h the length of the off periods remains constant in time, however, the number of blocks dropped during an off period increases. Consequently, there is no gain from additionally allocated resources during the off period itself, only after the off period the additional resources allow serving backlogged data faster. As a consequence, it can be concluded that a strict prioritization of a single flow or mobile terminal, which is at a significant disadvantage for other users, does in case of slowly fading channels not generally resolve quality of service problems. Fig. 4 compares delay bounds under opportunistic scheduling and processor sharing. The block error probability q/(p + q) is varied and at the same time the average channel rate h q/(p + q) = 3.6 is kept constant by adjusting the peak channel rate h accordingly. To set the remaining free parameter we choose p q =1/900. Delay bounds are derived for the periodic source according to Tab. I which competes for resources with an additional greedy source. In case of processor sharing the periodic source is assigned a channel with peak rate h/ and average rate h/ q/(p + q) =1.8. Under opportunistic scheduling the periodic source sees an onoff channel where the channel rate in on state is either h/ or h depending on whether the channel seen by the competing, greedy source is in on state or not. We start with an investigation of the stability region of the channel as seen by the periodic source. Under processor sharing the average channel rate as seen by the periodic source is h/ q/(p + q) =1.8. Hence, the stability region is upper bounded by 1.8. In contrast, under opportunistic scheduling the stability region increases with the block error probability from 1.8 up to 3.6, taking advantage from the opportunistic channel allocation where the probability that both sources see the channel in on state simultaneously decreases as the block error probability increases. Regarding the delay bounds, a significant impact of the block error probability can be noticed. With increasing block error probability the channel becomes bursty, it provides a higher rate when it is in on state, however, it is less frequently in on state, which degrades delay bounds significantly. With increasing channel burstiness the advantage of opportunistic scheduling in terms of the stability region becomes more and more apparent, however, the gain regarding the delay bound is comparably small as also obtained in [4]. As before increasing the channel rate, as effectively done by opportunistic scheduling, can only compensate marginally for adverse effects due to frequent and long off-periods.

6 V. CONCLUSIONS We showed a methodology for deriving service guarantees in radio communication systems. The key challenge in analyzing such systems are the temporal failures inherent in fading channels described by the Gilbert-Elliott model. To this end this work uses a probabilistic service curve approach, which facilitates the application of network calculus with MGFs to derive probabilistic backlog and delay bounds for radio links. Example applications comprise wireless video streaming where the presented framework supports determining the playout delay and dimensioning of the de-jitter buffer. A further important area of application is the derivation of end-to-end performance bounds for multi-hop wireless networks. APPENDIX: DERIVATION OF (6) From [10], [15], [14] the respective MGF follows as ( ) ([ ] )( ) λ M A (θ, t) = λ + µ, µ µ + θh µ 1 exp t. λ + µ λ λ 1 For the exponential of a real matrix it is known that ([ ]) a b exp = c d [ ] Φ cosh( )+ (a d) sinh( ) b sinh( ) c sinh( ) cosh( )+ (d a) sinh( ) where = (a d) +4bc/ and Φ=e (a+d)/ [3]. Thus M A (θ, t)= Φ ( ( λ cosh( ) + λ + µ + θh ) t sinh( ) λ + µ + µ ( cosh( ) + λ + µ θh )) t sinh( ) λ + µ =t (θh + λ µ) +4µλ/ and Φ=e t(θh λ µ)/. With cosh(x) =(e x +e x )/ and sinh(x) =(e x e x )/ and after some reordering it follows that ( (e + e ) + ( e e ) M A (θ, t) = Φ (λ + µ) +(λ µ)θh (λ + µ) (θh + λ µ) +4µλ ). (7) With λ, µ > 0 it can be shown that the fraction in (7) is smaller than one. For t 0 we have 0 and e e 0 and it holds that M A (θ, t) Φe = e t which completes the derivation. ( (λ µ+hθ) +4λµ λ µ+hθ) REFERENCES [1] R.-R. Boorstyn, A. Burchard, J. Liebeherr, and C. Oottamakorn. 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