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Modelling the guaranteed QoS for wireless sensor networks: a network calculus approach
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 82 (2011)
Abstract
Wireless sensor networks (WSNs) became one of the high technology domains during the last 10 years. Realtime applications for them make it necessary to provide the guaranteed quality of service (QoS). The main contributions of this article are a system skeleton and a guaranteed QoS model that are suitable for the WSNs. To do it, we develop a sensor node model based on virtual buffer sharing and present a twolayer scheduling model using the network calculus. With the system skeleton, we develop a guaranteed QoS model, such as the upper bounds on buffer queue length/delay/effective bandwidth, and singlehop/multihops delay/jitter/effective bandwidth. Numerical results show the system skeleton and the guaranteed QoS model are scalable for different types of flows, including the selfsimilar traffic flows, and the parameters of flow regulators and service curves of sensor nodes affect them. Our proposal leads to buffer dimensioning, guaranteed QoS support and control in the WSNs.
1. Introduction
Wireless sensor networks (WSNs) have been became one of the high technology domains of the seven seas, and theoretic and applications study about them are more and more regarded in recent years [1–3]. Realtime application areas for the WSNs encompass tracking, environment scouting, forecasting and medical care. Sink nodes of the WSNs respond in time on needs, so data channel between sink nodes and sensor nodes must offer a guaranteed quality of service (QoS). It includes deterministic sending rate, transmission without loss, endtoend delay with upper bound and so on [1]. The guaranteed QoS plays an important role in data transmission for the WSNs. For example, the endtoend delay with upper bound is one of the guaranteed services, whether the upper bound on endtoend can obtain a guarantee is a key to provide the guaranteed QoS and to complete effectively routing, congestion control and load balancing. To fulfill aims, the WSNs need to send some special probe packets [4]. The extra cost accounts for much total power under constrained energy, bandwidth and buffer size of a sensor node. However, it results in shortening of the WSNs' lifetime, and it is important to provide the guaranteed QoS model and the performance evaluation method for the WSNs.
Network calculus is a set of recent developments that enable the effective derivation of deterministic performance bounds in networking [5, 6]. Compared with some traditional statistic theories, network calculus has the merit that provides deep insights into performance analysis of deterministic bounds. Now, research areas for the network calculus include mostly QoS control, resource allocation and scheduling, and buffer/delay dimensioning in the virtual circuit switched networks, the guaranteed service networks and the aggregate scheduling networks [5].
In recent years, the endtoend delay bounds, in FIFOmultiplexing tandems, were estimated based on the least upper delay bound (LUDB) method [7]. The delay of individual traffic flows, in feedforward networks under arbitrary multiplexing, was computed [8]. The maximum endtoend delay, for a given flow in any feedforward network under blind multiplexing, was calculated [9]. Resource allocation and congestion control was investigated in distributed sensor networks using the network calculus [10]. An analytical framework was presented, based on the network calculus, to analyse worstcase performance and to dimension resources of sensor networks [11–14]. The power management problem in video sensor networks was investigated [15]. The worstcase performance of the WSNs was analysed [16]. Recently, the clustertree WSNs were modelled and dimensioned in the network calculus [17–19].
In previous studies [20–23], we drawn the deterministic performance bound on endtoend delay jitter for selfsimilar traffic regulated by a fractal leaky bucket regulator in a generalized processor sharing system, and obtained the deterministic and statistical performance bounds on endtoend delay in the WSNs and the wireless mesh networks.
In this article, we describe a generalized scenario of the WSNs, and present a practicable model of sensor nodes for guaranteed service support using a scheme based on virtual buffer sharing. On the basis of the notion of flows and microflows, we propose, using arrival curves and service curves in the network calculus, a twolayer scheduling model for sensor nodes. We develop a guaranteed QoS model, including the upper bounds on buffer queue length/delay/effective bandwidth, and singlehop/multihops delay/jitter/effective bandwidth. Combined with the research results of predecessor researchers, the main different contributions of this study are as follows. First, we present a system skeleton and a guaranteed QoS model that are suitable for the WSNs with some characteristics of distribution and multihops, and the sensor node model which not only fulfills these wants, but also makes performance analysis simpler. Second, we find that quantitative relations between the upper bounds on buffer queue length/delay/effective bandwidth, and singlehop/multihops delay/jitter/effective bandwidth and the service rate, the latency of the service curves in sensor nodes, and as well as the hops. Third, we reveal the impact of the service rate, the latency and the parameters of the regulators, including the Hurst parameter of selfsimilar traffic flows, on the guaranteed QoS. The findings' contributions are used to modelling the guaranteed QoS for the WSNs, and they may have potential applications to buffer and delay dimensioning, QoS support, routing implementing, congestion control and load balancing for the WSNs and other wireless networks with some characteristics of distribution and multihops.
The rest of the article is organized as follows. Section 2 devotes to the background knowledge of the network calculus. Section 3 discusses a system skeleton, including a generalized scenario of the WSNs, a sensor node model, the flow source model, the guaranteed QoS service and the scheduling model of a sensor node. Section 4 draws the upper bounds on the guaranteed QoS model. Section 5 shows the numerical results and compares one another to demonstrate the availability and the merits of the proposed skeleton, the guaranteed QoS model and our approach through same examples. Finally, Section 6 contains the summary of the results, some inferring remarks and future works.
2. Background on network calculus
In this section, we provide a brief background on the network calculus used in the article. Network calculus is the results of the studies on traffic flow problems, minplus algebra and maxplus algebra applied to qualitative or quantitative analysis for networks in recent years, and it belongs to tropical algebra and topical algebra.
Network calculus can be classified into two types: deterministic network calculus and statistical network calculus. The former, using arrival curves and service curves, is mainly used to obtain the exact solution of the bounds on network performance, such as queue length and queue delay, and so on. And then the latter, based on arrival curves and effective service curves, is used to obtain the stochastic or statistical bounds on the network performance. Here, we give only the necessary introductory material used in this article.
Theorem 1 (queue length and queue delay): Assume a flow passes through a sensor node, and the sensor node has an arrival curve α(t) and offers a service curve β(t). The queue length Q and the queue delay D of the flow, passing through the sensor node, satisfy, respectively,
and
Theorem 2 (multihops service curve): Assume a flow passes through the sensor node 1, node 2, ..., node N in sequence. Assume the sensor nodes offer the service curves of β^{(1)}, β^{(2)}, ..., β^{(N)}to the flow, respectively. The fixed delays between two neighbor sensor nodes are d_{1}, d_{2}, ..., d_{ N }_{1} in sequence. The multihops service curve β^{m}^{}^{h} satisfies
where ⊗ is the operator of the minplus convolution given by
and δ_{ d } is called a burst delay function. For 0 ≤ t ≤ d, δ_{ d } (t) = 0, and for t > d, δ_{ d } (t) = +∞.
In Equation 3, we obtain, setting n = 2, the singlehop service curve β^{s}^{}^{h} as follows
The proof of the theorems and more information about the network calculus are found in [5, 6].
3. System skeleton
3.1. System model
In the following, we firstly describe a generalized scenario of the WSNs, where includes sink nodes, sensor nodes and a sensor field as shown in Figure 1.
When certain sensor node of the sensor field probes an occurring event, the sensor node sends probed data to one of its neighbor sensor nodes according to the route arithmetic arranged in advance, and then the neighbor sensor node sends the data to one of its neighbor sensor nodes. Finally, the data probed by the first sensor node is transmitted to a sink node passing multihops.
In general, the energy of a sensor node is supplied by battery under constrained energy, so the storage and communication capacity of a sensor node is constrained. It is essential to provide the guaranteed QoS to lessen spending and to prolong a network lifetime.
The next, we present, using a scheme based on virtual buffer sharing, a sensor node model as shown in Figure 2. The buffer of the sensor node is allocated to data channels between the sensor node and its upstream neighbor nodes. The probed data from its upstream neighbor nodes share the buffer of the sensor node. The scheduler of the sensor node sends the data to the downstream neighbor nodes according to the QoS priority. Figure 2 shows the case for the sensor node j and i upstream neighbor nodes, including the sensor node 1, node 2, ..., node i.
Remark 1: The sensor node model using the virtual buffer sharing has some merits as follows.

(1)
The model provides a minimum guaranteed service rate for every data channel from upstream neighbor nodes under constrained bandwidth, namely, when the data flow passes through a sensor node, the node guarantees a minimum service rate.

(2)
The buffer and the bandwidth of a sensor node are shared by all of upstream neighbor nodes and delivered to them in part to their weights, so the WSNs obtain a larger gain from the statistical multiplexing of independent flows.

(3)
The model makes performance analysis simpler, and it is suitable for mobile sensor nodes in the WSNs.
3.2. Flow source model
The dynamic and complexity properties of the network and the fluctuation of the traffic possibly cause the burstiness of the traffic flows in the WSNs. They increase the average delay and result in the unfairness of resource allocation. It becomes more difficult in providing or analysing the guaranteed QoS. In this article, we can categorize traffic flows into two types: flow and microflow. The former contains file flows, audio flows and video flows and so on. The latter, belonging to the identical type, aggregates a flow. The aggregate flow enters a sharing buffer to queue and schedule for the sensor node. In this article, we select the leaky bucket source model due to its simplicity and practical applicability, and use leaky bucket regulators to regulate the microflows at every sensor node, to enable nonrule microflows to be restraint under the certain conditions. The microflow, regulated by the leaky bucket regulator, is indicated by envelope α(t) as shown in Equation 4,
where the case of M = 1 agrees to the simple leaky bucket regulator, b is interpreted as the burst parameter, and r as the average arrival rate.
Remark 2: the microflow in an interval [t, t+τ] is denoted by A(t, t+τ), and it has the following properties as shown in [24].
Property 1 (additivity):
Property 2 (subadditive bound):
Property 3 (independence): all microflows are independent.
3.3. Guaranteed QoS
The guaranteed QoS provides the QoS guarantees which involve the stability of performance, the usability and reliability of calculation resources, as well as the rationality of calculation price. In this article, we mainly discuss how to provide guarantees for the QoS, including the upper bounds on buffer queue length/delay/effective bandwidth, and the upper bounds on singlehop/multihops delay/jitter/effective bandwidth. It is important to limit the values of buffer queue length/delay/jitter to a sustainable level below the upper bounds. For example, once the value of tracking or environment scouting delay is beyond a certain value, such as the upper bound on endtoend delay in the WSNs, the accuracy of tracking and the effectiveness of environment scouting have sharply declined. Table 1 reports an example of guaranteed service that comes from the experimental results for a realtime tracking environment and scouting application in the clustertree WSNs based on IEEE 802.15.4/ZigBee protocol in [19].
3.4. Twolayer scheduling model
In the following, we present a twolayer scheduling model of a sensor node as shown in Figure 3. The process of the model is as follows: First, the microflows, entering a sensor node, with the same or similar QoS are regulated by a leaky bucket regulator given in Equation 4, and serves for the arrival curve α(t) of the next buffer. The functions α(t) and A(t, t+τ) satisfy Property 2; Second, we assume that the first come, first served strategy is adopted in the buffer, and the microflows, belonging to the same type, enter a special buffer assigned by the sensor node; Finally, the aggregate flows are scheduled in a way of a service curve β(t). The service curve is shown as follows.
From Properties 1 and 3, and Figure 3, the aggregate flows A_{ j } (t, t+τ) and microflows A_{ j,k } (t, t+τ), k = 1,2,...,n satisfy
From [25], the equivalent envelope curve α_{ j } (t) of the aggregate flows and the envelope curve α_{ j,k } (t) (k = 1,2,...,n) of the microflows satisfy
The service curve β_{ i } (t) of the flow i is defined as
where β(t) is interpreted as a service curve of sensor node, α_{ k } as an arrival curve of the buffer k, and n as the number of the buffers in the sensor node.
In order to simplify the calculation, without loss of generality, we assume the service curve β(t) of the sensor node is a ratelatency function β_{ R,T } (t) given by
where R is interpreted as the service rate, T as the latency. Obviously for R > 0 and 0 ≤ t ≤ T, we have β_{ R,T } (t) = 0.
From Property 3, Equations 5 and 6, the simple leaky bucket regulator is used, and the envelope curve of the regulator is
From Equations 4 and 8, if ${\sum}_{i=1}^{n}{r}_{i}<R$, then the parameter θ_{ i } is optimized, and we have
Substituting θ_{ i } into Equation 7, and combining Equation 6 with Equation 9, we obtain
Remark 3: From Equation 10, we have known, each flow, which enters the sensor node scheduler, holds a certain service curve, and the service curve will not only be decided by the total service curve of the sensor node scheduler, but also by the arrival curve of the flow.
4. Guaranteed QoS model
In this section, we present, using the network calculus, the guaranteed QoS model. The model is mainly used in two aspects: one is the offline dimensioning of a system, which is responsible for the quantification to obtain the prearranged resources providing the guaranteed QoS; and the other is the online admission control, which is responsible to decide whether receives a new flow according to the QoS requirements and the usable resources. In the following, the guaranteed QoS model, including the upper bounds on Q_{ i } , D_{ i } , e_{ i } , DD_{ N } , ΔD_{ N } and ee_{ N } (in Table 2) of the system skeleton in Section 3 are discussed in the network calculus.
4.1. Node QoS model
Proposition 1: (upper bound on buffer queue length): In an interval [0, t], the upper bound on Q_{ i } satisfies
Proof: From Equation 1, we have
Substituting Equations 9 and 10 into Equation 12, we hold
Proposition 2: (upper bound on buffer queue delay): In an interval [0, t], the upper bound on D_{ i } satisfies
Proof: From Equation 2, we obtain
Substituting Equations 9 and 10 into Equation 14, we have
For $R\ge {\sum}_{k=1}^{n}{\sum}_{j=1}^{{c}_{j}}{r}_{k,j}$, from Equation 15, we obtain
Proposition 3: (upper bound on buffer effective bandwidth): In an interval [0, t], the upper bound on e_{ i } satisfies
where D_{ i } is given by Equation 13.
Proof: Substituting Equation 9 into Equation 1.30 in [5], we have Equation 16.
Remark 4: The leaky bucket regulators and aggregators do not increase the upper bounds on buffer queue length/delay/effective bandwidth of a sensor node, and also do not increase the buffer requirements of the sensor node.
4.2. Singlehop and multihops QoS model
Proposition 4: (upper bound on singlehop and multihops delay): Assume a flow passes through the sensor node 1, node 2, ..., node N in sequence, and the sensor node i offers the service curves of β^{(1)}, β^{(2)}, ..., β^{(N)}to the flow, respectively. The fixed delays between two neighbor sensor nodes are d_{1}, d_{2}, ..., d_{ N }_{1} in sequence. The upper bound on DD_{ N } satisfies
and
where R_{ i } and T_{ i } are interpreted as the service rate and the latency of the sensor node i, and ${r}_{k,j}^{\left(i\right)}$ and ${b}_{k,j}^{\left(i\right)}$ as the burst parameter and the average arrival rate of the leaky bucket regulator of the sensor node i, respectively.
Proof: From Equations 10, 3 and 8, we hold
Substituting Equations 9 and 18 into Equation 2, we have Equation 17.
Proposition 5 (upper bound on singlehop and multihops delay jitter): Assume a flow passes through the sensor node 1, node 2, ..., node N in sequence, and the sensor node i offers the service curves of β^{(1)}, β^{(2)}, ..., β^{(N)}to the flow, respectively. The fixed delays between two neighbor sensor nodes are d_{1}, d_{2}, ..., d_{ N }_{1} in sequence. The upper bound on ΔD_{ N } satisfies
where T_{1} is interpreted as the latency of the first sensor node, ${b}_{i,k}^{\left(1\right)}$ as the burst parameter of the microflow k of the flow i, entering the first sensor node, and others in Equation 19 are shown in Equation 17.
Proof: The upper bound on DD_{ N } obtained from Equation 17 is the total delay, and the upper bound on ΔD_{ N } and the fixed delay D_{ c } hold ΔD_{ N } = DD_{ N }  D_{ c } . The multihops fixed delay is defined as ${D}_{c}={\sum}_{i=1}^{N1}{d}_{i}$. Therefore, Equation 19 exists obviously.
Proposition 6 (upper bound on singlehop and multihops effective bandwidth): Assume a flow passes through the sensor node 1, node 2, ..., node N in sequence, and the sensor node i offers the service curves of β^{(1)}, β^{(2)}, ..., β^{(N)}to the flow, respectively. The fixed delays between two neighbor sensor nodes are d_{1}, d_{2}, ..., d_{ N }_{1} in sequence. The upper bound on ee_{ N } satisfies
where the parameters in Equation 20 are given in Equation 19.
Proof: From Equations 9 and 16, we obtain
for D_{ i,k } ≥ D_{ i } , from Equations 17 and 16, we have Equation 20.
Remark 5: The singlehop scenario is a special case of the multihops WSNs. In Equations 17, 19 and 20, we obtain the singlehop QoS model for N = 2, and obtain the multihops QoS model for N > 2.
Remark 6: The leaky bucket regulators and aggregators do not increase the upper bounds on singlehop/multihops delay/jitter/effective bandwidth of the WSNs.
5. Numerical results
In this section, we give the numerical results to demonstrate the effectiveness and the simplicity of our method. Without loss of generality, we research a general scenario of the WSNs as shown in Figure 4. If N = 2, then there is a singlehop case, otherwise, there is a multihops case. The twolayer scheduling model presented in Section 3 is used for all sensor nodes. The service curves β(t) of the sensor nodes are given in Equation 10, where R is interpreted as the service rate and T as the latency of the service curves of the sensor nodes. The fixed delay between two neighbor sensor nodes is marked as d.
Figure 4 shows the transmission of three flows in the WSN. The three flows, namely, flow1, flow2 and flow3, are marked as A_{1}(t), A_{2}(t) and A_{3}(t), respectively. They come from the sensor nodes A, B and C. Hence, without any loss of generality, we assume the flow A_{1}(t) contains three microflows: A_{1,1}(t), A_{1,2}(t), A_{1,3}(t), the flow A_{2}(t) contains two microflows: A_{2,1}(t) and A_{2,2}(t), and the flow A_{3}(t) contains one microflow: A_{3,1}(t).
Recent research suggests that the sensory data flow is bounded by arrival curve α(t) = 576(bps) + 390(b) x t in the clustertree WSNs based on IEEE 802.15.4/ZigBee protocol in [19]. Here, we consider the case of M = 2 in Equation 4 and assume that every microflow is regulated by the leaky bucket regulator α(t) as shown in Equation 9. The average arrival rate r_{ i,k } and the burst tolerance b_{ i,k } of the six microflows are shown in Table 3. Obviously, the arrival curves of the flows are given by Equation 10.
Remark 7: The units of buffer queue length Q, effective bandwidth e and ee are Mb, the units of delay D and DD, the time t, the latency T and the fixed delay d are ms and the unit of the service rate R is Mbps except the units that are given.
5.1. Node QoS
In the following, we discuss the relations between the sensor node QoS and the parameters of the service curve provided by the sensor nodes, and the time evolution of the sensor node QoS.
Figure 5 shows the impact of the service rate R and the latency T on the upper bounds on buffer queue length Q and the evolution laws of Q in a sensor node. We see a straightforward dependency: the upper bound on Q is smaller for smaller service rate R with lowvalue or smaller latency T; it is smaller for larger service rate R with highvalue or larger evolution time t. For all flows, the changing tendency of the upper bound on Q with the increase of the service rate R or the latency T and the time evolution t of Q are the same. The size deviation of the upper bounds on Q_{1}, Q_{2} and Q_{3} of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is equal regardless of R values and T values. The upper bound on Q_{2} of A_{2}(t) is more than that of Q_{1} of A_{1}(t), and that of Q_{3} of the A_{3}(t) is smallest. Obviously, the impart of the latency T or burst tolerance b on the upper bound on Q is more than that of the service rate R or the average arrival rate r, respectively.
Figure 5a plots the Q curves as a function of the service rate R. The upper bound on Q for any flow reaches a maximum Q_{max} when the service rate R = 128, and the Q_{max} value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 1.81, 2.03 and 1.53, respectively. The shapes of curves at the two sides of the maximum point are asymmetric. For example, at the distance 50 from the maximum point on the left, the Q_{max} value of the three flows is 1.81, 2.02 and 1.52, but the Q_{max} value of the three flows is 1.81, 2.03 and 1.53, respectively, at the same distance on the right.
Figure 5b plots the Q curves as a function of the latency T. The upper bound on Q increases linearly with the increase of the latency T, and the slope of each line is 9.764 × 10^{4}.
Figure 5c plots the Q curves as a function of the evolution time t. There exists a linear relationship between the upper bound on Q and the evolution time t and the same slopes of the all lines are 9.642 × 10^{4}.
Figure 6 shows the impact of the service rate R and the latency T on the upper bounds on buffer queue delay D in a sensor node. We see a straightforward dependency: the upper bound on D is smaller for larger service rate R; it is smaller for smaller latency T.
Figure 6a plots the D curves as a function of the service rate R. The D values, curving inwards, decay with the increase of R regardless of T values, nearly converging 0 for all flows. The decay rates in the upper bounds on D by the near exponential increase with the increase of the service rate R for certain flow, and increase with the increase of the burst tolerance b of the flows with the same service rate T. For instance, if T = 1 and R = 50, then the D value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 38.7, 43.3 and 32.8, and if T = 1 and R = 200, then the D value of the three flows is 10.3, 11.4 and 8.9, respectively.
Figure 6b plots the D curves as a function of the latency T. The upper bounds on D increase linearly with the increase of the latency T regardless of the R values. The slopes of all lines are 1.
Figure 7 shows the impact of the service rate R and the latency T on the upper bound on buffer effective bandwidth e in a sensor node. We see a straightforward dependency: the upper bound on e is larger for larger service rate R; it is larger for smaller latency T.
Figure 7a plots an e curve as a function of the service rate R. The e values increase with the increase of R values, and the increase rate is getting smaller and smaller with the increase of R values for certain flow regardless of the values of the latency T. The delay rates of the increase rates decrease with the increase of the burst tolerance b of the flows. For instance, if T = 1 and R = 50, then the e value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 12.41, 16.17 and 6.10, and if T = 1 and R = 200, then the e value of the three flows is 46.47, 61.18 and 22.44, respectively.
Figure 7b plots an e curve as a function of the latency T. The upper bounds on e decrease with the increase of T values, and the decay rate is getting smaller and smaller with the increase of T values for certain flow regardless of the R values. The e curves of all flows are near parallel.
In summary, the performance curves denote the upper bounds of the sensor node QoS. In Figures 5, 6 and 7, the curves show the deterministic worstcase length/delay/effective bandwidth in the buffer queue of a sensor node, respectively. It means that the values of the buffer queue length/delay must are lower than the values of the performance curves. We can reduce, regulating the average arrival rate r and the burst tolerance b of the microflows by controlling the parameters of the regulators or regulating the service rate R or the latency T of a sensor node by controlling the parameters of the scheduler, the values of the upper bounds on buffer queue length/delay of a sensor node to achieve these purposes that the buffer queue length/delay is very small. Instead, we can increase, regulating the average arrival rate r and the burst tolerance b or regulating the service rate R or the latency T, the value of the upper bound on buffer queue effective bandwidth to obtain a guaranteed bandwidth for those flows through the sensor node, and eventually reduce the buffer queue delay.
5.2. Multihops and singlehop QoS
In the following, we discuss the relations between the multihops QoS and the singlehop QoS and the parameters of the service curve provided by the sensor nodes and the hops. We still use the general scenario of WSNs as known in Figure 4.
5.2.1. The case 1
The sensor nodes (node 1, node 2, ..., node N1, node N) have the same service curves: β_{1}(t) = β_{2}(t) = ... = β_{ N }_{1}(t) = β_{ N } (t) = β(t) = R(t  T). From Equation 8, we have R_{1} = R_{2} = ... = R_{ N }_{1} = R_{ N } = R, and T_{1} = T_{2} = ... = T_{ N }_{1} = T_{ N } = T. To make easy the following discussion, we assume that the fixed delays between two neighbor sensor nodes are the same: d_{1} = d_{2} = ... = d_{ N }_{1} = d. First, we investigate the multihops scenario with hops higher than 2.
Figure 8 shows the impact of the service rate R, the latency T and the hops N on the upper bounds on multihops delay DD. We see a straightforward dependency: the upper bound on DD is smaller for larger service rate R; it is smaller for smaller latency T and smaller hops N.
Figure 8a plots a DD curves as a function of the service rate R. The DD values, curving inwards, decay with the increase of R regardless of T values, N values and d values for certain flow. The decay rates in the upper bounds on DD by the near exponential increase with the increase of the service rate R for certain flow, and increase slightly with the increase of the burst tolerance b of the flows for the same T. For instance, if T_{1} = T_{2} = ... = T_{ N }_{1} = T_{ N } = T = 1, R_{1} = R_{2} = ... = R_{ N }_{1} = R_{ N } = R = 50, d_{1} = d_{2} = ... = d_{N1} = d = 2 and N = 10, the DD value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 315, 320 and 309, and if T = 1, R = 200, d = 2, and N = 10, the DD value of the three flows is 100, 102 and 99, respectively.
Figure 8b plots a DD curves as a function of the latency T. The upper bounds on DD increase in linear with the increasing of T values regardless of R values, N values and d values for certain flow. All the increase rates of DD are 11.
Figure 8c plots a DD curves as a function of the hops N. The upper bounds on DD increase in linear with the increase of N regardless of R values, T values and d values for certain flow. All the increase rates of DD are 0.017.
Remark 8: From Equations 17 and 19, we have the relation between the multihops delay jitter ΔD and the multihops delay DD as follows: ΔD = DD  Σd, where d is the fixed delay between two neighbor sensor nodes. As a result, we can obtain some numerical results about the upper bounds on ΔD by setting d_{1} = d_{2} = ... = d_{ N }_{1} = d = 0, and the impact of the service rate R, the latency T and the hops N on ΔD is similar to those on DD.
Figure 9 shows the impact of the service rate R, the latency T and the hops N on the upper bounds on multihops effective bandwidth ee. We see a straightforward dependency: the upper bound on ee is larger for larger service rate; it is larger for smaller latency and smaller hops.
Figure 9a plots an ee curves as a function of the service rate R. The upper bounds on ee increase with the increase of R values, and the increase rate is getting smaller and smaller with the increase of R for certain flow regardless of the values of the latency T, the fixed delay d and the hops N. The impact of the burst tolerance b on ee is more than that of the service rate R on ee for the highvalues R > 30 or the impact of the service rate R is more. For example, if N = 10, T = 1, R = 20 and d = 2, the ee value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 1.32, 1.26 and 0.30, and if N = 10, T = 1, R = 200 and d = 2, the ee value of the three flows is 4.98, 6.89 and 2.02, respectively.
Figure 9b plots an ee curves as a function of the latency T. The upper bounds on ee decrease with the increase of T values, and the decay rate is getting smaller and smaller with the increase of T for certain flow regardless of the values of the service rate R, the fixed delay d and the hops N. The changing tendency of ee for each flow is similar to that of e.
Figure 9c plots an ee curves as a function of the hops N. The upper bounds on ee decrease with the increase of N values. The decay rates of ee by the near exponential increase with the increase of the hops N for all flows, and they are smaller for larger burst tolerance b of the flows. For instance, if N = 1, R = 100, T = 1 and d = 2, then the ee value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 23.17, 30.48 and 11.21, and if N = 5, R = 100, T = 1 and d = 2, the ee value of the three flows is 56.19, 77.63 and 23.52, respectively.
In the case 1, assuming N = 2, we can obtain the singlehop QoS. The study result shows the service rate R and the latency T produce the same impact on the upper bounds on singlehop delay DD and the multihops delay DD, and the singlehop effective bandwidth ee and the multihops effective bandwidth ee. If R = 100 and T = 1 and d = 2, the upper bounds on singlehop delay DD of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) are 0.021, 0.023 and 0.018, and the upper bounds on singlehop effective bandwidth ee are 23.2, 30.5, and 11.2, respectively.
To summarize, the performance curves denote the upper bounds of the singlehop/multihops QoS. In Figures 8 and 9, the curves show the deterministic worstcase endtoend delay/effective bandwidth. It means that the values of the endtoend delay must are lower than the values of the performance curves. We can reduce, by regulating the average arrival rate r and the burst tolerance b or the service rate R and the latency T of all sensor nodes on an endtoend path, the values of the upper bounds on endtoend delay to achieve this purpose that the endtoend delay/jitter is very small. On the other side, we can increase, by regulating the average arrival rate r and the burst tolerance b or the service rate R and the latency T, the value of the upper bound on endtoend effective bandwidth to gain a guaranteed bandwidth for those flows through the endtoend path, and eventually reduce the endtoend delay/jitter.
5.2.2. The case 2
The sensor nodes (node 1, node 2, ..., node N1, node N), given in Figure 4, have the different service curves: β_{1}(t) ≠ β_{2}(t) ≠ ... ≠ β_{ N }_{1}(t) ≠ β_{ N } (t). By the number of the flows and the values of the average arrival rate and the burst tolerance of the arrival curves as shown in Table 3 without any loss of generality, we assume that the parameters of the service curves of the five sensor nodes (node 1, node 2, node 3, node 4, node 5), used for numerical calculation in the following, are given in Table 4.
Next, we calculate the upper bounds on multihops delay DD, the multihops delay jitter ΔD and the multihops effective bandwidth ee of the three flows (Table 3) from the sensor node 1 to the sensor node 5. If the fixed delay between two neighbor sensor nodes d_{1} = 1.2, d_{2} = 2.3, d_{3} = 2.0, d_{4} = 3.5, d_{5} = 2.6, from Equations 17, 19 and 20, then we hold the DD value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 58.9, 59.4 and 58.2, and the ΔD value of the three flows is 47.3, 47.8 and 46.6, and the ee value of the three flows is 8.15, 11.78 and 3.43, respectively.
The case 2 is the general form of the case 1. Using the same method, we calculate the upper bound on singlehop delay/jitter/effective bandwidth. The singlehop QoS is compared to the multihops QoS, and shows similar trend between them. But, unlike the case 1, we should consider how to regulate the various service rate R and the various latency T of every sensor node on an endtoend path. The aim is to reduce the values of the upper bounds on endtoend delay/effective bandwidth, and obtain the tolerable delay/jitter for the tracking and environment scouting applications in the WSNs.
5.2.3. The case 3
To display, the guaranteed QoS model appears to, presented in Sections 3 and 4, be valid for the WSNs with selfsimilar traffic flows.
The method is as follows: we replace the simple leaky bucket regulators with fractal leaky bucket regulators in the twolayer scheduling model of sensor nodes; the envelope of the fractal leaky bucket regulators is also expressed as shown in Equation 12 in [26], and the average arrival rate r_{ i,k } and the burst tolerance b_{ i,k } are interpreted as follows, respectively,
where m_{ i,k } is interpreted as the longterm average arrival rate of selfsimilar traffic, σ_{ i,k } as the standard deviation, H_{ i,k } as the Hurst parameter with the values ranging from 0.5 to 1, and γ is a positive constant of 6.
Remark 9: The fractal leaky bucket regulators and aggregators do not increase the upper bounds on buffer queue length/delay and the buffer requirements of a sensor node, and do not increase the upper bounds on singlehop/multihops delay/jitter/effective bandwidth of the WSNs.
To provide comparisons between performances with selfsimilar traffic flows and that with general traffic flows for the greater details in the research analysis, we consider the general scenario of the WSNs shown in Figure 4and three selfsimilar traffic flows including six selfsimilar microflows. Without loss of generality, for any i and k, we assume that the m_{ i,k } and σ_{ i,k } values of the selfsimilar microflows are equal to the r_{ i,k } and b_{ i,k } values of the microflows in Table 3, respectively.
Figure 10 shows the impact of the Hurst parameter H on the upper bounds on multihops delay DD and multihops effective bandwidth ee in case of the selfsimilar microflows with the same Hurst parameters. We see a straightforward dependency: the upper bound on DD and ee are smaller for larger Hurst parameter.
The DD values and the ee values decrease with the increasing of the H values under an increasing rate. The impact of the Hurst parameter on the DD values and the ee values increase with the increase of the standard deviation σ_{ i,k } . For instance, if H = 0.75, R = 100, T = 1, d = 2 and N = 10, then the DD and ee value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 215.9, 218.9 and 212.1, and 3.95, 5.26 and 1.55, and if H = 0.95 and the same values of other parameters mentioned above, the DD value and the ee value of the three flows are 129, 131 and 127, and 2.34, 2.82 and 0.83, respectively.
Now, we assume that the Hurst parameters of the selfsimilar microflows have different values. For example, if H_{1,1} = 0.90, H_{1,2} = 0.80, H_{1,3} = 0.75, H_{2,1} = 0.85, H_{2,2} = 0.60 and H_{3,1} = 0.70, and R = 100, T = 1, d = 2 and N = 10, then the DD value and the ee value of the flows: A_{1}(t), A_{2}(t) and A_{3}(t) is 222, 226 and 218, and 3.76, 5.74 and 1.65, respectively.
Besides, calculation and analysis of the node QoS and the singlehop QoS, such as the upper bounds on buffer queue length/delay/effective bandwidth of a sensor node, and the singlehop delay/jitter/effective bandwidth, and the multihops delay jitter in the WSNs with selfsimilar traffic flows are done, and the results are similar.
In this special case with selfsimilar traffic flows, we can obtain the guaranteed QoS by regulating the service rate R and the latency T of sensor nodes, as we can do in the case 2. The difference is that we use the fractal leaky bucket regulators to regulate the arrival selfsimilar traffic flows. Obviously getting the Hurst parameter of the arrival selfsimilar traffic flow is a key in advance in the case. Then, we can obtain the guaranteed QoS by regulating the average arrival rate r and the burst tolerance b of selfsimilar traffic flows in the WSNs.
Remark 10: Recently, many related studies on deterministic endtoend delays have done. Lenzini et al. [7] computed the endtoend delay based on the LUDB methodology, but the delay is the minimum among all the delay bounds and the LUDB methodology cannot be applied directly to nonnested tandems. Schmitt et al. [8] achieved the worstcase endtoend delays under blind multiplexing in tandem networks, and they dealt with arrival and service curves by a decomposition and recomposition scheme. Unlike Schmitt's method, Bouillard et al. [9] directly computed the worstcase endtoend delay instead of looking first for an endtoend service curve by a decomposition and recomposition scheme, and may obtain a tighter bounds and a cheap complexity. Koubaa et al. [17–19] proposed closedform recurrent expressions for computing the worstcase endtoend delays across any sourcedestination path in a clustertree WSN. In this article, the proposed networkcalculusbased models are simpler for computing the upper bounds on buffer queue length/delay/effective bandwidth and multihops/singlehop delay/jitter/effective bandwidth using virtual buffer sharing in the WSNs, and these models are suitable for various flows, including selfsimilar traffic flows.
In summary, based on the numerical results and analysis, we have found that the parameters of the flow regulators and the service curves in the sensor nodes play an important role in modelling on a guaranteed QoS model for the WSNs, and have obtained the following findings: (1) the upper bound on buffer queue length is smaller for larger service rate with highvalues, and it is smaller for larger evolution time or larger Hurst parameter, and it is smaller for smaller latency or smaller service rate with lowvalues; (2) the upper bound on (multihops/singlehop) delay/jitter is smaller for larger service rate or larger Hurst parameter, and it is smaller for smaller latency or smaller hops; (3) the upper bound on (multihops/singlehop) effective bandwidth is larger for larger service rate, and it is smaller for larger latency or larger hops or larger Hurst parameter. In order to obtain network performance optimization and the guaranteed QoS of the WSNs, including low delay for tracking, we can reduce the upper bounds on (endtoend) delay/jitter or increase the upper bounds on (endtoend) effective bandwidth by designing the rational regulator parameters, including the average arrival rate and the burst tolerance, and the rational scheduler parameters such as the service rate and the latency, of sensor nodes.
6. Conclusion
In this article, we have discussed the problem of the guaranteed QoS for flows. First, based on the arrival curve and the service curve in the network calculus, we have presented the system skeleton, involving the sensor node model on virtual buffer sharing, the flow source model and the twolayer scheduling model of sensor nodes and so on. Second, with the system skeleton, we have not only drawn the node QoS model, such as the upper bounds on buffer queue length/delay/effective bandwidth, but also drawn the singlehop/multihops QoS model, such as the upper bounds on singlehop/multihops delay/jitter/effective bandwidth. Finally, we have shown the practicability and the simplicity of the model and our approach using example results in the article. We can optimize network performances by designing reasonable regulators and schedulers of the WSNs nodes. A network calculus approach is as a tradeoff between complexity and accuracy. It is general, simple and practicable for provisioning the guaranteed QoS in the WSNs and other wireless networks with some characteristics of the distribution and the multihops.
Ongoing and future works include: (1) implementing the algorithmic build upon the proposed networkcalculusbased model to ensure polynomial time complexity, for example, the computational complexity of node QoS algorithmic is O(cn), where c is the number of microflows, n is the number of flows, and the computational complexity of buffer queue length algorithmic is O(c^{2}n^{2}), and the computational complexity of multihops/singlehop QoS algorithmic is O(cnN), where N is the number of hops; (2) investigating statistical sensor network calculus in order to capture the stochastic and dynamic behaviors of the WSNs.
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Acknowledgements
This research was supported in part by the grant from the National Natural Science Foundation of China (60973129, 60903058 and 60903168), the China Postdoctoral Science Foundation funded project (200902324), the Specialized Research Fund for the Doctoral Program of Higher Education (200805331109), the Scientific Research Fund of Hunan Provincial Education Department of China (10B062), the Program for Excellent Talents in Hunan Normal University (ET10902 and ET51102) and the Startup Project for Doctoral Research Supported by Scientific Research Fund of Hunan Normal University (110608). The material in this article was presented in part at First International Conference on Communications and Networking in China (ChinaCOM'06), Beijing, China, October 2527, 2006, the International Workshop of Information Technology and Security (WITS'08), Shanghai, China, December 2022, 2008, and the ISECS International Colloquium on Computing, Communication, Control, and Management (CCCM'08), Guangzhou, China, August 0405, 2008. The authors would like to thank the reviewers for their valuable comments.
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Zhang, L., Yu, J. & Deng, X. Modelling the guaranteed QoS for wireless sensor networks: a network calculus approach. J Wireless Com Network 2011, 82 (2011). https://doi.org/10.1186/16871499201182
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Keywords
 wireless sensor networks
 quality of service
 network calculus
 upper bounds