I Introduction
Services utilizing communications between machines are expected to receive a lot of attention, such as health monitoring, security monitoring and smart grid services [1]. These Internet of Things (IoT) services generate new demands for wireless networks. The spectrum of service scenarios in the IoT is wide and as a result the required quality of service (QoS) across IoT services is also wide. In some scenarios ultra high reliability is required, in others a low latency is required and supporting massive numbers of lowcost and lowcomplexity devices is still important issue. The devices can be served by the cellular networks and, specifically, by their M2Mevolved versions, such as Narrowband IoT (NBIoT) [2]. However, there is a lowcost alternative for serving these devices using Low Power Wide Area (LPWA) networks that operate in unlicensed bands. The number of IoT devices connected by noncellular technologies is expected to grow by 10 billions from 2015 to 2021 [3]. It is therefore of interest to develop QoS models for the LPWA protocols in order to analyze which protocol is best suited for a given service.
Long Range Widearea Network (LoRaWAN) is an emerging protocol for lowcomplexity wireless communication in the unlicensed spectrum using Long Range (LoRa) modulation. The scalability and capacity of LoRaWAN is investigated in [4] where it is implicitly assumed that the interarrival times are fixed. In [5] the scalability is evaluated in terms of goodput and network energy consumption. One of the key elements of LoRaWAN is the use of duty cycling in order to comply with the requirements for unlicensed operation. Duty cycling is imposed per subband by regulation and optionally also aggregated for all bands. It is the central factor that sets limitation on the throughput and the latency of the network. The limits of dutycycled LoRaWAN are pointed out in [6], but only aggregated duty cycle and fixed interarrival arrivals are considered.
The contribution of this paper is an analytical model of the LoRaWAN uplink (UL) that characterizes the performance, in terms of latency and collision rate, under the influence of regulatory and aggregated duty cycling, assuming exponential interarrival times. The obtained latency and collision rate results from the analysis are verified through simulation.
Ii Long Range Wide Area Network
LoRaWAN is a wireless communication protocol providing long range connectivity at a low bit rate. LoRaWAN is based on the LoRa modulation. LoRaWAN supports LoRa spreading factors 7 to 12. The overhead of a LoRaWAN message with a payload and no optional MAC command included is 13 bytes.
LoRaWAN defines a MAC layer protocol to enable low power wide area networks (LPWAN) [7]. A gateway serves multiple devices in a star topology and relays messages to a central server. LoRaWAN implements an adaptive data rate (ADR) scheme, which allows a network server to select both the data rate and the channels to be used by each node.
Three different classes (A, B and C) of nodes are defined in LoRaWAN. Class A has the lowest complexity and energy usage. All LoRaWAN devices must implement the class A capability. A class A device can receive downlink messages only in a receive window. There are two receive windows after a transmission in the uplink. The first window is scheduled to open 1 to 15 second(s) after the end of an uplink transmission with a negligible 20 ms margin of error. The second window opens 1 second after the end of the first.
LoRaWAN utilizes the industrial, scientific and medical (ISM) radio bands, which are unlicensed and subject to regulations in terms of maximum transmit power, duty cycle and bandwidth. The enddevice also obeys a duty cycling mechanism called the aggregated duty cycle, which limits the radio emission of the device. An aggregated duty cycle of 100 % corresponds to the device being allowed to transmit at any time, but still in accordance with the regulatory duty cycling. The lowest aggregated duty cycle of 0 % means that the particular device turns off the transmissions completely.
Iii System Model
Consider devices connected to a single LoRaWAN gateway. Each device is assigned a spreading factor to use for transmission by a network server. We account for the interference through the collision model, where collision occurs when two or more devices try to transmit simultaneously in the same channel using the same spreading factor. We also consider a LoRaonly configuration, in this work, such that no interference from other technologies is present. Different spreading factors are considered to be entirely orthogonal. A fixed payload size is assumed. We further assume that all devices are class A and have successfully joined the network and transmit the messages without acknowledgement so that there are no downlink transmissions. Due to the absence of acknowledgements, retransmissions are not considered.
Among all subbands, a device is given a subset of the subbands. Enumerate these subbands 1 through . Let , and , be the number of channels and the dutycycle^{1}^{1}1 is a normalized value between [0, 1]. in subband , respectively. As described in the specifications [7] and in the source code of the reference implementation of a LoRa/LoRaWAN device^{2}^{2}2https://github.com/Loranet, the scheduling of a LoRaWAN transmission happens as follows:

A device waits until the end of any receive window.

A device waits for any offperiod due to aggregated duty cycling.

A device checks for available subbands, i.e., ones that are not unavailable due to regulatory duty cycling:

A channel is selected uniformly randomly from the set of channels in all available subbands.

If there is no free subband, the transmission is queued in the first free subband. A random channel in that subband will be selected.

A transmission, limited by the duty cycle , with a transmission period infers a holding period, which, including the transmission itself is given by:
(1) 
The service rate is the inverse of the holding time, . Subbands can have different duty cycles and in turn different service rates. Let be the generation rate of packets for a device. When several subbands are defined for the device the subband for the next transmission is selected according to the step 3a) and 3b). We define service ratio as the fraction of transmissions carried out in the th subband.
Iv Analytical Model
In this section the analytical models for latency and collision probability are presented.
Iva Single Device Model: Latency
The latency of a transmission is the time spent on processing, queueing, transmission of symbols, and propagation. Assuming that the time for processing and propagation are negligible, we have:
(2) 
We model the wait for reception windows and aggregated duty cycling ( steps 1) and 2) ) as a single traffic shaping queue. The service rate of this queue is the slowest mean rate of service in step 1) and 2). For step 3), we model the regulatory duty cycling as an queue with heterogeneous servers, where each server corresponds to a subband. The waiting time for a transmission and the service ratio of each subband can then be found from queue theory.
The waiting time, , due to regulatory dutycycling can be calculated for asymmetric queue^{3}^{3}3The term “asymmetric” captures the heterogeneous service rates of subbands. that models step 3) of the scheduling procedure, but as it is easier to model and compute on a queue relative to a queue, we use the rule of thumb that the waiting line of a symmetric queue is approximately twice that of an queue [8] to simplify our analysis. Our simulations show that this is a good approximation also for asymmetric queues.
The waiting time in subband is then:
(3) 
where is the ErlangC probability that all servers are busy. The transmission latency in each band can then be found from Eq. (2). The mean latency is given as a weighted sum of the transmission latencies in each subband, where the weights are given by the service rate of each subband.
The fraction of transmissions in subband , , is the product of the holdingefficiency of the subband (fraction of time it is held) and the service rate of the band throughout that period. Then the service ratio is:
(4) 
where is the probability that the subband is idle.
The service ratio can be expressed in shorthand forms for the two extreme cases of all subbands being available or busy all of the time. When all subbands are available at the time of a transmissionthe channel of transmission is selected uniformly from the set of all channels as per step 3a):
(5) 
In the case that all subbands are unavailable at the time of a transmissionthe transmission is carried out in the next available subband as per step 3b):
(6) 
In order to describe between these extremes, we must find
. Hence, we wish to find the steadystate probabilities given a Markov model of the subband selection behaviour. For this purpose the model of a
jockeying^{4}^{4}4Jockeying: A packet changes queue to a shorter queue if, upon the end of service of another packet, it is located in a longer queue. queue from [9] has been adopted. The Markov model of the jockeying queue has a limited state space since, by definition, the difference in the number of queued transmissions in any two subbands may not be larger than one. This allows us to put up a matrix containing all state transition probabilities, which can be used to evaluate the steady state probabilities, , by solving the linear system . It also allows adoption of a Markov model for LoRaWAN device behaviour, which is step 3) in the subband selection, by introducing state transition probabilities based on the number of channels in each nonbusy subband in .The jockeying queue does has a limited state space. As in [9] we approximate the model by making it finite by limiting the queue sizes to 1000. The model now allows us to compute the steady state probabilities of all states; Amongst them and . Then and can be calculated from Eq. (3) and Eq. (4). Note that waiting times are lower for a jockeying queue than a regular queue. Hence applying the rule of thump for approximation of an queue from a queue on a queue, will yield a lower latency approximation of the queue.
IvB Multiple Devices Model: Collisions
In this work we assume that no devices are making use of the optional acknowledgement feature of LoRaWAN. Hence there is no DL in the model and as another consequence no retransmissions occur upon collision.
It is empirically found in [10] that spreading factors are not orthogonal in practice and, due to capture effect, one transmission may be received successfully if the power of the wanted transmissions is sufficiently greater than the interfering one. Unfortunately, at present there is no model of capture effect in LoRaWAN and in this work, for simplicity, we assume that all channels and all SFs are orthogonal. When two or more transmissions happen in the same channel, using the same SF, at the same time, they collide. This means we can model the access scheme as multichannel ALOHA random access, as in [4, 6, 11]. Since there are 6 spreading factors defined for LoRaWAN, we have 6 sets of orthogonal Alohachannels in subband . The collision rate must be evaluated for each spreading factor.
We found the service ratios of each subband in Section IVA. Since the number of devices, the transmission time for the spreading factor being evaluated and the mean inter arrival time are known, we can calculate the load within a subband. The load within the subband is spread uniformly over the channels allocated to that band. Hence the traffic load of M devices, in subband i, given is
(7) 
where is the percentage of all devices , which use the ’th spreading factor in subband .
The collision probability is then
(8) 
In the paper, only unacknowledged UL transmissions are considered. So DL limitations and retransmissions are not considered in this work. Therefore the outage is caused by collisions can be quantified by our model.
V Performance Evaluation
In this section the latency given by Eq. (2), the service ratios given by Eq. (4) and the collision probability given by Eq. (8) are evaluated numerically. The evaluation is done for SF 12 based on 125 kHz channels, 50 bytes payload, 13 bytes overhead, code rate 4 and preamble length .
The latency including the transmission time and the waiting time due to regulatory duty cycling as a function of arrival rate are depicted in Fig. 1. The latency is plotted for standalone usage of each subband (G to G4) and for two subband combinations (G+G1 and G+G2).
The analytical approximation using Eq. (3) for a heterogeneous queue provides a tight upper bound of the cases for the multiple subbands (G+G1 and G+G2) and a tight approximation for the single band cases. The latency obtained by the jockeying queue provides a lower approximation. The results show that lower latencies and higher capacities can be achieved for subbands with higher dutycycles and combinations of bands with high dutycycles.
The service ratios for the cases with combined subbands are plotted in Fig. 2. We see that combining G with G1 and G2, respectively, leads to very different service ratios for the bands. G contains 15 channels and G1 contains just 3, but they have the same dutycycle. The combination of G and G1 yields the service ratio limit for G for low arrival rates, but since the dutycycling is the same for the subbands we have the limit for a high arrival rate.
The consequence of the subband pairing becomes evident by the collision rates depicted in Fig. 3. We see that the collision rate for G+G1 is larger than that of G alone or G+G2. This is due to the traffic not being spread equally on the channels for high arrival rates for G+G1. Since the limits of G+G2 are much closer, the load is spread more uniformly over the channels at high arrival rates and we see a drop in collision rate by adding the subband. Notice that the devices reach their capacities before the collision rate comes close to 1. In this way dutycycling limits the collision rate for each band, allowing for more devices to share the band, but in practice arrivals beyond the capacity of each device would be dropped.
From Fig. 1 it seems that the subband with the highest dutycycle, G4, is attractive as it delivers low latency even at very high loads. However, when collisions are taken into account, we see that the subband has a very high collision rate as it only contains a single subchannel. On the other hand, the lowest dutycycle is found in subband G2, which has relatively high latency even at low loads, but with a lower collision rate than G4.
In Fig. 4 the service ratios for G+G4 with an aggregate duty cycle of 0.05 (equivalent to a service rate capacity of the queue is 0.0146) and an aggregate duty cycle of 0.075 (equivalent to 0.0219) are plotted. The introduction of the aggregated duty cycle ( queue) was found to effect the regulatory duty cycle queue ( queue) by the service capacity, which freezes the subband service ratios of the regulatory queue and limits the obtainable latency.
Vi Concluding Remarks
A model for evaluating the performance of LoRaWAN UL in terms of latency and collision probability was presented. The numerical evaluation was done for EU868 ISM band regulations, but the analysis is also valid for other bands utilizing duty cycling, such as the CN779787 ISM band.
Shorthand forms for the limits of were presented. Equalizing the limits keeps the collision rate of subband combining at a minimum. The tradeoff for this is a higher latency. The traffic shaping effect of aggregated dutycycling was shown and may be used as a builtin tool for collisionlatency tradeoff when combining subbands.
The UL model presented in this work, can be combined with DL models for Class A, B and C LoRaWAN devices and more sophisticated collision models to give insight into the bidirectional performance in LoRaWAN.
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