A Appendices

Optimizing Data Aggregation for Uplink Machine-to-Machine Communication Networks


Machine-to-machine (M2M) communication’s severe power limitations challenge the interconnectivity, access management, and reliable communication of data. In densely deployed M2M networks, controlling and aggregating the generated data is critical. We propose an energy efficient data aggregation scheme for a hierarchical M2M network. We develop a coverage probability-based optimal data aggregation scheme for M2M devices to minimize the average total energy expenditure per unit area per unit time or simply the energy density of an M2M communication network. Our analysis exposes the key tradeoffs between the energy density of the M2M network and the coverage characteristics for successive and parallel transmission schemes that can be either half-duplex or full-duplex. Comparing the rate and energy performances of the transmission models, we observe that successive mode and half-duplex parallel mode have better coverage characteristics compared to full-duplex parallel scheme. Simulation results show that the uplink coverage characteristics dominate the trend of the energy consumption for both successive and parallel schemes.

1 Introduction

Machine-to-machine (M2M) applications are rapidly growing, and will become an increasingly important source of traffic and revenue on 4G and 5G cellular networks. Unlike video applications, which are expected to consume around 70% of all wireless data by the end of the decade [1], M2M devices will use a comparatively small fraction. However, M2M communication has its own challenges. The air interface design for high-data-rate applications may not effectively support M2M’s vast number of devices, each usually having only a small amount of data to transmit. Thus, M2M will require sophisticated access management and resource allocation with QoS constraints to prevent debilitating random access channel (RACH) congestion [2], and to enable link adaptation with low overhead, reduced energy consumption, and efficient control channel design [3]. Bluetooth (IEEE 802.15.1), Zigbee (IEEE 802.15.4) and WiFi (IEEE 802.11b) are a few examples of current technologies that have been used for M2M communication [3]. Meanwhile, 3GPP has been studying M2M communication in its standardization process for Long Term Evolution-Advanced (LTE-A). In Release 13, a low-complexity user equipment (UE) category for M2M devices is specified, in which a UE has reduced bandwidth and lower maximum transmission power, and can operate in all the duplex modes [4]. However, there is still no consensus on the general network architecture for large scale M2M communication.

1.1 Motivation and Related Work

Unlike most human generated or consumed traffic, M2M as defined in this paper is characterized by a very large number of small transactions, often from battery powered devices. The power and energy optimal uplink design for various access strategies is studied in [5], while an optimal uncoordinated strategy to maximize the average throughput for a time-slotted RACH is developed in [6]. For the small payload sizes relevant for M2M, a strategy that transmits both identity and data over the RACH is shown to support significantly more devices compared to the conventional approach, where transmissions are scheduled after an initial random-access stage.

Clustering is a key technique to reduce energy consumption and it increases the scalability and lifetime of the network [7, 8]. A Low-Energy Adaptive Clustering Hierarchy protocol (LEACH) to evenly distribute the energy load among the sensors and enable scalability for dynamic networks by incorporating localized coordination is proposed [9, 10]. A low overhead protocol that periodically selects cluster-heads according to the node residual energy to support scalable data aggregation is established in [11]. A Distributed Energy-Efficient Clustering (DEEC) scheme, where the cluster-heads are elected by a probability based on the ratio between residual energy of each node and the average energy of the network is proposed in [12]. A data gathering mechanism, where clusters closer to the BS have smaller sizes than those farther away from the BS to balance the energy consumption is analyzed in [13]. Although these techniques are energy efficient, they do not incorporate the coverage characteristics of wireless transmissions.

A stochastic analysis to determine how many data collectors per area are required to meet the outage probability for a given sensor node density is implemented in [14]. Modulation strategies to minimize the total energy to send a given number of bits are analyzed in [15]. The energy efficiency of a large scale interference limited ad hoc network at physical, medium access control (MAC) and routing layers is quantified [16]. Optimal power allocation to minimize the total energy consumption in a clustered WSN where sensors cooperatively relay data to nearby clusters is studied in [17]. The energy-optimal relay distance and the optimal energy to transmit a bit successfully over a particular distance are analyzed in [18].

Different control mechanisms to avoid congestion caused by random channel access of M2M devices are reviewed [19]. An adaptive slotted ALOHA scheme for random access control of M2M systems with bursty traffic that achieves near-optimal access delay performance is proposed [20]. A comprehensive performance evaluation of the energy efficiency of the random access mechanism of LTE and its role for M2M is provided [21]. An energy-efficient uplink design for LTE networks in M2M and human-to-human coexistence scenarios that satisfies quality-of-service (QoS) requirements is developed [22]. Similar to [21, 22], we study an energy-efficient design for M2M uplink where devices perform multi-hop transmissions. We also incorporate the coverage characteristics for different transmission modes using stochastic geometry.

Because low-power M2M devices may not be able to communicate with the BS directly, hierarchical architectures may be necessary. Hence, critical design issues also include optimizing hierarchical organization of the devices and energy efficient data aggregation. Although these issues have not been studied in the context of M2M, there is prior work on distributed networks in the context of wired communications. In [23], energy consumption is optimized by studying a distributed protocol for stationary ad hoc networks. In [24], a distribution problem which consists of subscribers, distribution and concentration points for a wired network model is studied to minimize the cost by optimizing the density of distribution points. In [25], a hierarchical network including sinks, aggregators and sensors is proposed, which yields significant energy savings.

Hierarchical networks can provide efficient data aggregation in M2M or other power-limited systems to enable successful end-to-end transmission. Despite previous research efforts, e.g., [14], [8] and [18], to the best of our knowledge, there has been no study focusing on data aggregation schemes for M2M networks together with the rate coverage characteristics, especially from an energy optimal design perspective. Providing such a study is the main contribution of this paper.

1.2 Contributions and Organization

A large-scale hierarchical wireless network model for M2M. We propose a new communication model for the M2M uplink. In Sect. 2, we consider a hierarchical scenario to model the wireless transmissions where the hierarchical levels describe the multi-hop stages of the model. Each level is composed of the transmitter and aggregator processes. Once the transmissions of the current hierarchical level are completed, the transmitters are either turned off or kept on depending on the transmission scheme used, and in the subsequent hierarchical level, the aggregator process forms the new transmitter process. Since each device can act as a transmitter or an aggregator in a particular time slot, we model the devices as wireless transceivers. In Sect. 4, the aggregation process is repeated over multiple levels to generate a hierarchical model. We show there is an optimal fraction of aggregators that minimizes the overall energy consumption.

SIR and rate coverage probability. In Sect. 5, we analyze the SIR coverage and rate coverage characteristics of the multi-stage transmission process to determine the optimal number of stages. We consider two possible transmission techniques: i) successive scheme, where the hierarchical levels are not active simultaneously, and ii) parallel scheme, where either all the levels are active simultaneously as in full-duplex transmission or where the active levels are interleaved as in half-duplex mode. We describe the proposed transmission models in detail in Sect. 6.

Optimizing the number of multi-hop stages. We propose a general aggregator model for power limited M2M devices. Since the M2M devices are power limited and thus range limited, multi-hop routing is a feasible strategy rather than direct transmissions. However, in designing multi-hop protocols, the number of hops cannot be increased arbitrarily due to the additional energy consumption incurred by relays; long-hop routing is a competitive strategy for many networks [26]. We find an upper and lower bound on the optimal number of hops in Sect. 5.

Optimizing the network energy density. We use a generic transceiver scheme to model the energy consumption of the devices. For multi-hop transmission models, we find the optimal fraction of aggregators independently chosen from the set of devices that minimizes the average total energy consumption per unit area, i.e., the mean total energy density, for a fixed payload per device of an M2M network. We incorporate the energy consumption of aggregators and transceiver circuit components into the energy model. In Sects. 3 and 4, we optimize the total energy density of the transmissions based on a given SIR coverage requirement for the devices.

We evaluate the performance of the proposed techniques in Sect. 7, and provide a comparison in terms of their energy densities and communication rates through numerical investigations.

2 System Model

We consider a cellular-based uplink model for M2M communication where the BS and device locations are distributed as independent Poisson Point Processes (PPPs)1 with respective densities of and with . Each BS has an average coverage area of , and each device has a fixed payload of bits to be transmitted to the BS. We also assume open loop power control with maximum transmit power constraint under which the transmit power of a device located at distance from the BS is2 , where is the path loss exponent, is the maximum transmit power constraint and is the received power when . With this assumption, the average received power at the BS from a device in its coverage area and located at distance from the BS is constant and equal to . Assuming devices3 are scheduled based on TDMA, the uplink SINR for any device is , where and are the intra cell and out of cell interferences4, is the power spectral density and is the bandwidth.

2.1 Data Aggregation and Transmission Model

Devices transmit data to the BS by aggregating data. The initial device process is independently thinned by probability to generate5 the aggregator process with density and the device (transmitter) process with density , where . Each transmitter is associated to the closest receiving device (aggregator), i.e., the transmitting devices within the Voronoi cell of the typical aggregator device will transmit their payloads to that aggregator.

The aggregation process can be extended to multiple stages. Each hierarchical level is composed of the transmitter and the aggregator processes, where the aggregator processes of all stages are initially determined such that they are disjoint from each other. At each stage, after the set of transmitters transmit their payloads to their nearest aggregators, and once the transmissions of a hierarchical level are completed, the transmitters are turned off and excluded from the process. In the subsequent hierarchical level, the aggregators of the previous stage become the transmitters, and they transmit their data to the aggregator process of the new stage. The aggregation process is repeated over multiple stages to generate the hierarchical transmission model. The process ends when all the payload is transmitted to the BS in the last stage of this multi-hop process, which we call a transmission cycle. The aggregation model will be detailed in Sects. 3 and 4.

2.2 Wireless Transceiver Model

We model the devices as wireless transceivers, since each device can act as a transmitter or an aggregator in a particular time slot. In [28], a generic architecture for energy-limited wireless transceivers is provided. The transceiver has four major building blocks. The transmit block (TX) is responsible for modulation and up-conversion, the receive block (RX) for down-conversion and demodulation, the local oscillator (LO) block for the generation of the required carrier frequency, and the power amplifier (PA) block for amplification of the signal to produce the required RF transmit power , where we assume the maximum transmit power is bounded and given by . The power consumption in the receive and transmit paths are


where the power consumption of , and blocks are denoted by , and , respectively, which are non negative constants. is the receiver overhead power that is assumed to be constant. The PA power consumption is given by , where is the PA efficiency, which is constant in the linear regime. The average energy cost of the transceiver is given by where and , and and stand for the average receive and transmit times. The symbol definitions are given in Table 1.

In the proposed model, multiple devices (transmitters) send data to an aggregator (receiver), where each device is allocated a different time slot on the same frequency, i.e., TDMA. Therefore, and for transmitting devices, and their energy cost is proportional to . For the aggregator, and , and its energy cost is proportional to . Let denote the average energy required by an aggregator. denotes the average energy required for all transmissions within the coverage of an aggregator node, i.e., if devices transmit to the aggregator, is the sum of their average transmission energies. Hence, for a total transmission/reception time slot of duration , aggregator and total transmitter energy consumptions are


where we assume devices sequentially transmit to the aggregator, the time slot satisfies , and we assume that the communication delay due to processing of the data is negligible. Hence, transmitted data can be received within the time slot allocated and the aggregator can decode the received data. We also assume that the transmissions and receptions are synchronized. is mainly determined by the reception duration, while depends on the energy consumption of the PA, and . Furthermore, whenever a device is in sleep mode, its transmitter and receiver modules are not active but other components, i.e., and blocks, are still consuming energy that justifies the scaled term of in (2).

We will use this transceiver model and the data aggregation strategy described to formulate our energy optimization problems in Sects. 3-4, and analyze the SIR coverage in Sect. 5.

Parameter Symbol Parameter Symbol
Power consumption in the receive (transmit) path () RF transmit power; maximum transmit power ;
Power consumption of the receive (transmit) block () RF receive power; mean total received power ;
Power consumption of the local oscillator (LO) Path loss exponent
Receiver overhead power Average receive (transmit) time in one cycle ()
Power consumption of the power amplifier PA Transmission/reception time slot duration
PA efficiency Receiver (transmitter) activity factors ()

Table 1: Notation.

3 Single-Stage Energy Density Optimization

The main focus of this section is to model the average total energy density for a single-stage data aggregation scheme, which is a single hop energy model incorporating the data aggregation and transceiver model described in Sect. 2. This model paves the way for understanding the multi-stage energy model to be discussed in Sect. 4.

For the single-stage model, recall that initially a density of the devices will be independently selected as aggregators, and a density of will be the transmitters. Then, the set of aggregators, , collects the data from the remaining devices, , based on nearest aggregator association, and each aggregator might have multiple devices assigned to it. However, multiple transmissions to an aggregator at a particular time slot are not allowed, hence the devices are scheduled based on TDMA. Once all devices complete their transmissions, the aggregators also incorporate their payloads, and then transmit the whole data to the BS.

We assume perfect channel inversion power control, under which the received power at the BS from any device is unity. Then, the average total uplink power is given by Theorem 1.

Theorem 1.

The mean total uplink power of the devices in the Voronoi cell of the typical aggregator, i.e., the average of the total power consumption of the PAs, is given by


where is defined as the critical distance.


Proof is given in Appendix A.1, where the final result can be obtained by incorporating the maximum power constraint into the mean additive characteristic associated with the typical cell of the access network model provided in [29, Ch. 4.5]. ∎

Corollary 1.

If the maximum transmit power is unbounded, the mean total uplink power becomes

Let be a random variable that denotes the number of devices within the Voronoi cell of a typical aggregator and its average can be obtained as


where is the expectation with respect to the Palm probability conditioned on [29].

We denote the number of devices within the Voronoi cell of a typical aggregator and within distance by , and its mean can be obtained as


where is the closed ball centered at and of radius .

Lemma 1.

The mean total received power is given by


Proof is given in Appendix A.2. ∎

We now define the average energy terms as functions of the aggregator and device densities.

Definition 1.

Average total energy in a Voronoi cell of an aggregator. This is the average total energy consumption due to the transmissions within the coverage of the typical aggregator for a fixed payload per device. The average total energy in a typical cell is given by

Definition 2.

Average total energy density. This is the average total energy consumption per unit area per unit time for a fixed payload per device, and can be found by scaling the average energy density per Voronoi cell by the number of Voronoi cells per area. The average number of Voronoi cells per unit area is given by .

The dissipated energy density, i.e., the energy consumption per unit area, is defined as


Our objective is to find the optimal value of that minimizes the total energy consumption per unit area. As increases, the total number of aggregators and their total energy consumption to receive data will increase, and as decreases, the distance between the aggregators and the energy requirement of the typical transmitter will increase. We modify (2) as


where we recall and is the second moment of the number of devices per aggregator6.

Any aggregator device aggregates data from multiple devices. Hence, the average number of devices per aggregator, i.e., , is always greater than 1. Thus, the optimal fraction of aggregators that minimizes the overall energy density should satisfy .

4 Multi-Stage Energy Density Optimization

We extend the model in Sect. 3 to the case where the aggregation process is repeated times before the data is eventually transmitted to the BS. Let and be the average energies required for reception and transmission in the coverage of an aggregator and be the aggregator density at stage . The average total energy density at stage is


The initial total density of the devices is given by . The initial process with density is independently thinned to obtain the set of aggregator processes with density for stage to form a disjoint set of aggregators for each stage. The density of aggregators at stage is denoted by , for . The total density of active devices that can be either transmitter and receiver at stage is given by .

The transmitter device density of the first stage is found by subtracting the total density of aggregators from the initial density of the device process as . By the end of the first stage, the devices with a density of will transmit their payload to the aggregator process with a density of . Next, at the second stage, the aggregators of the first stage form the new transmitting device process, i.e., , and these devices transmit to the devices forming the set of second stage aggregator process with density . The aggregation process can be extended to stages in a similar manner by letting for . In Fig. 1, we illustrate a three-stage model, where the aggregator process of stage , i.e., , is obtained by independently thinning with probability , i.e., has a density of .

Figure 1: A three-stage aggregation scheme with PPP() device process. The aggregator processes (stage 1: , stage 2: , stage 3: ) are obtained by thinning the original PPP, and have densities for stages k=1:3. Density for the initial transmitter process (stage 1: ) is , and for the transmitter processes for later stages are for , respectively. Last stage aggregators correspond to the BSs.

Generalizing (9), the average energy consumptions per unit area at stage are given by


where , and is the joint SIR coverage probability up to and including stage at a given threshold. SIR coverage probability gives the fraction of successful transmissions with respect to an SIR threshold, and the actual average energy at stage is determined by the fraction of successful transmissions up to stage , i.e., the joint SIR coverage . Hence, the current stage energy consumption scales with the fraction of the successful transmissions. For the devices in outage, no additional energy expenditure is incurred in the consecutive stages. In this paper, we interchangeably use (or ), (or ), (or ), and (or ) when the stage index is insignificant or clear from context.

4.1 An Upper Bound on the Energy Density

Let . Following the aggregation procedure, an upper bound for the average total energy consumption density over stages is found by ignoring the SIR coverage probability as that is equivalent to


where . We simplify (13) by letting as this scaling does not affect the result of the optimization formulation.

The slot duration at stage is and the duration that the transmit block is on is . The relation between the average transmit times are as . The transmit duration at slot is proportional to the mean total number of bits to be transmitted as . Note that ’s for are dependent random variables because of the correlation between the Voronoi cell areas of the aggregators at subsequent stages [31]. In fact, the inequality follows from the observation that ’s are positively associated, i.e., when the values of one variable tend to increase as the values of the other variable increase as can be seen from Fig. 2.

Figure 2: Numerical correlation over multiple stages.

We denote the correlation between the number of devices within the Voronoi cells of the typical aggregators of the consecutive stages and , i.e., the correlation between and , by , which is expressed as


To the best of our knowledge, there is no analytical model in the literature for the correlation between the number of stages for the proposed data aggregation model. However, we can observe the trend of the correlation numerically. We illustrate how the correlation behaves for number of stages in Fig. 2. From Table 2, the average number of devices per aggregator is for stage 1 and for stages . As increases, decreases with a rate which is faster than for with a decrease rate of . Therefore, dominates the trend of and thus, we observe a different behavior for compared to for , where the correlation is similar because the device and aggregator densities for higher stages are obtained with the same scaling parameter . Correlation satisfies for , which is sufficiently small given large number of stages. Thus, letting for analytical tractability, and assuming inter-stage independence among ’s, we can modify the expectation of the product as

Remark 1.

If there are stages in total, at the last stage, all the aggregators of the previous stage (stage ) with density are transmitting to the BSs with density . Thus, at stage , . Therefore, the fraction of aggregators at the last stage is


which is incorporated into the numerical analysis to find the minimizing aggregator fraction.

4.2 Average Energy Density with an SIR Coverage Constraint

This section mainly concentrates on the average energy consumption in the case of SIR outage. The average energy density of the proposed aggregator model is found by incorporating the SIR coverage probability of the typical transmitting device. If the received SIR at any stage cannot exceed the threshold , then the transmission is unsuccessful. Therefore, devices with a density of will be successful at stage , i.e., out of transmitting devices served per aggregator, on average, only of them will successfully transmit their data, and the rest of the devices will not meet the minimum SIR requirement despite consuming energy. Therefore, the performance of the current stage depends on the previous stages.

Incorporating the SIR coverage, the average total energy density in (13) is modified as


where denotes the joint SIR coverage probability for stages and , where we do not consider the effect of SIR coverage probability as all the devices of first stage transmit even though they might not be successful. In Sect. 5, we consider sequential and parallel modes with the inter-stage independence assumption, i.e., the joint SIR coverage becomes , where the dependence on is inherent and omitted to keep the notation concise.

Next, we formulate the energy density optimization problem assuming that SIR coverage probability of each device is unity, i.e., for . Note that depends on the communication scheme, the network model parameters, such as device density, fading distribution, and the path loss, and is determined independently from the energy model. This section mainly concentrates on the energy model, while the SIR coverage and rate models – and hence, ’s for various transmission schemes – will be discussed extensively in Sects. 5-6. In Sect. 7, we combine the energy results with the coverage characteristics for the successive and parallel modes, to evaluate the energy efficiencies of the proposed models.

4.3 Energy Density Optimization Problem

The following optimization problem minimizes the average total energy density over stages:


where the aggregator density is for , and for stage .

For the optimization formulation in (18), and for , we let . The important design parameters of the total energy density are tabulated in Table 2. Using (3) and the design parameters in Table 2, the mean total uplink power for different stages is given by

Proposition 1.

Combining (17) with the constraints in (18) for unit SIR coverage probability, we define to be the total energy density at stage . Hence, is given by


Incorporating the constraints, and using (20), the objective function of (18) is equivalent to


Similarly, the energy upper bound can be minimized by evaluating (21) at , and taking its derivative with respect to .

Proposition 2.

is an increasing function of for , and is a decreasing function of total number of stages .


Note that increases with , and is an increasing function of for . Using (20), it is easy to show that for for fixed . Note also that decreases, and does not change with , but decreases with , hence it is trivial to show that is decreasing in . ∎

Proposition 2 implies that the energy density is higher for higher order stages. On the other hand, as the number of stages increases, the energy density at the last stage decreases. The number of stages cannot be increased arbitrarily. Next, to capture the energy consumption tradeoff between the stages, we investigate the effect of on the total energy density.

Remark 2.

Using (20), we can easily observe that is decreasing in for . Note also that is given by the following sum

We observe that the first term decays with as decreases with , and is invariant to . The second term is not strictly a decreasing function of . On the other hand, it is strictly increasing for and , where and decreasing in . As the second term is scaled by , which is typically very high for the operating regime of M2M, it determines the trend of total energy.

Table 2: Design parameters.
Proposition 3.

Let be the minimizer of the energy optimization problem in (21) with the assumption7 of , and is increasing in . Then, is a decreasing function of the total number of stages .


See Appendix A.3. ∎

This section concentrates on the energy efficiency of M2M assuming perfect coverage. We now study the SIR coverage of the hierarchical M2M model for different transmission schemes.

5 SIR Coverage Probability

The data aggregation models in Sects. 3 and 4 do not incorporate the fact that the transmissions are not always successful. In this section, we generalize this to a coverage-based model, where the transmission is successful if the SIR of the device is above a threshold. We derive the probability of coverage in the uplink for the proposed data aggregation model in Sects. 3 and 4. We assume that the tagged aggregator and tagged devices experience Rayleigh fading, and we ignore shadowing. The transmit power at the typical node at distance from its aggregator is , and the received power is , where .

Orthogonal access is assumed in the uplink and at any given resource block, there is at most one device transmitting in each cell. Let be the point process denoting the location of devices transmitting on the same resource as the typical device. The uplink SIR of the typical device located at distance from its associated BS (aggregator) on a given resource block is


where and denote the distance between the transmitter aggregator pair and the distance between the interferer and the typical aggregator, respectively. The random variable is the small-scale iid fading parameter due to interferer .

Assumption 1.

The actual distribution of is very hard to characterize due to the randomness both in the area of the Voronoi cell of the aggregator and in the number of the devices it serves. Therefore, we approximate it by the distance of a randomly chosen point in to its closest aggregator and hence it can be approximated by a Rayleigh distribution [32]:


The uplink SIR coverage of the proposed system model is given by the following Lemma.

Lemma 2.

The uplink SIR coverage with truncated power control: With truncated power control and with minimum average path loss association8, the uplink SIR coverage is given by


where , is Rayleigh distributed with parameter , and


denotes the Laplace transform of the interference where , and is the Gauss-Hypergeometric function.


See Appendix A.4. ∎

Corollary 2.

The uplink SIR coverage with open loop power control [33]: With open loop power control and with minimum average path loss association, the uplink SIR coverage is


As , and , which yields the final result. ∎

Corollary 3.

The uplink SIR coverage is lower bounded by


where is a lower bound for the Laplace transform of the interference and given as


Noting that for , we obtain the following upper bound for the conditional expectation in step (g) of (56):


where the RHS can be calculated as


From (57), (29) and (30), we can lower bound , which yields the final result. ∎

5.1 Coverage Probability and Number of Stages

The number of multi-hop stages is mainly determined by the SIR coverage and the distance coverage, which we define as the probability that the distance between a device and its nearest aggregator is below a threshold. We provide bounds on using these coverage concepts.

Assumption 2.

Interstage independence. The proposed hierarchical aggregation model introduces dependence among the stages of multi-hop communication since each subsequent stage is generated by the thinning of the previous stage. For analytical tractability, we assume that the multi-hop stages are independent of each other9. Hence, the transmission is successful if and only if all the individual stages are successful. The success probability over stages is

where denotes the coverage probability at stage . With full channel inversion and minimum average path loss association, the uplink SIR coverage is independent of the infrastructure density as given in (26). For the case , since in (26) is also independent of the device density, and only depends on the threshold and path loss exponent , is identical for all stages, and denoted by .

Lemma 3.

An upper bound on K (Sequential mode). Given a minimum probability of coverage requirement and an SIR threshold , the number of stages is upper bounded by


The upper bound is obtained by combining (24) with the condition and using the relation