I Introduction
Virtual reality (VR) is often seen as one of the most important applications in 5G cellular systems [ABIQualcommVR:17, EjderVR:17, ParkWCL:18]. As in real life, mobile VR users can interact in the virtual space immersively with virtual objects that may stimulate their multiple sensory organs. This multimodal VR perception happens, for example, when a VR user measures the size of a virtual object through visual and haptic senses. In this study, we consider the problem of supporting such visuohaptic VR perceptions over wireless cellular networks, and focus on the downlink design.
The key challenge is that these two perceptions have completely different cellular service requirements. In fact, visual traffic requires high data rate and relatively low reliability with packet error rate (PER) on the order of [Shi:10, UR2Cspaswin:17]. This requirements can be supported mostly through enhanced mobile broadband (eMBB) links [ITU5G:15]. Haptic traffic, by contrast, should guarantee a fixed target rate and high reliability with PER on the order of [Steinbach:12, Zhang:18], which can be satisfied via ultrareliable and low latency communication (URLLC) links [PetarURLLC:17, MehdiURLLC:18].
Furthermore, in order to render a smooth multimodal experience, the PERs associated with the visuohaptic VR perceptions should guarantee a target perceptual resolution. To be precise, the perceptual resolution is commonly measured by using the justnoticeable difference (JND) in psychophysics, a field of study that focuses on the quantitative relation between physical stimulus and perception [Ernst:2002aa, Shi:10, ShiHirche:16]. Following Weber’s law, JND describes the minimum detectable change amount of perceptual inputs, e.g., mm for the object size measurement using visuohaptic perceptions [Ernst:2002aa]
. According to psychophysical experiments, the JND of the aggregate visuohaptic perception is the harmonic mean of the squared JNDs of the individual perceptions
[Ernst:2002aa], in which the JND of each perception is proportional to the PER [Shi:10].As a result, the PERs associated with the visuohaptic VR traffic should be adjusted so as to achieve a target JND, while abiding by the eMBB and URLLC service objectives in terms of PERs and data rates. Due to the discrepancy of the visual and haptic service requirements, it is difficult to support both perceptions through either eMBB or URLLC links. Hence, it is necessary to slice the visuohaptic VR traffic into eMBB and URLLC links, leading to URLLCeMBB multimodal transmissions. Unfortunately, such multimodal transmissions bring about multimodal selfinterference, which is manifested through an actual wireless interference as visualized in Figs. 1a and b, or via the necessity to share resources as shown in Figs. 1b and c.
This critical selfinterference can be alleviated by multiplexing the URLLCeMBB multimodal transmissions over the transmit power domain with successive interference cancellation (SIC) at reception, i.e., downlink nonorthogonal multiple access (NOMA) [3GPPMUST:2015], as illustrated in Fig. 1b. Alternatively, as Fig. 1c shows, the selfinterference can be avoided via orthogonal multiple access (OMA) such as frequency division multiple access (FDMA). In this paper, using stochastic geometry, we investigate the optimal design of NOMA and OMA to support visuohaptic VR perceptions while coping with the multimodal selfinterference in a largescale downlink system.
Related Works – The communication and computation resource management of mobile VR networks has recently been investigated in [EjderVR:17, OsvaldoAR:17, ChenSaad:17, ParkWCL:18, Elbamby:18], particularly under a VR social network application [ParkWCL:18] and a VR gaming scenario [Elbamby:18]. The endtoend latency has been studied in [OsvaldoAR:17] for a singlecell scenario and in [ChenSaad:17, ParkWCL:18] for a multicell scenario. These works focus primarily on supporting either visual or haptic perceptions. Towards supporting multimodal perceptions, suitable network architecture and coding design have been proposed in [Zhang:18, Steinbach:12], while not specifying the requirements on the wireless links. In an uplink singlecell system, orthogonal/nonorthogonal multiplexing of URLLC and eMBB links has been optimized by exploiting their reliability diversity in [Petar5G:18].
Contributions – The main contributions of this work are summarized as follows.

To the best of our knowledge, this is the first work that combines both visual and haptic modalities in the context of mobile VR network design.

To support visuohaptic VR perceptions, an optimal downlink NOMA design with reliabilityordered SIC has been proposed (see Lemma 2 and Proposition 4).

Compared to an OMA baseline (see Proposition 2), it has been observed that the proposed NOMA becomes preferable under a higher target integratedperceptual resolution and/or a higher target rate for haptic perceptions (see Fig. 3).

By using stochastic geometry, closedform average rate expressions have been derived for downlink URLLCeMBB multiplexing under OMA and NOMA in a largescale cellular network (see Propositions 1 and 3).
Ii System Model and Problem Formulation
In this section, we first introduce the downlink system operation of OMA and NOMA under a singlecell scenario, and describe its extension to the operation under a largescale network. Then, we specify visuohaptic perceptions, followed by the problem formulation of visuohaptic VR traffic slicing and multiplexing.
The user under study requests visuohaptic VR perceptions that are supported through URLLCeMBB cellular links. We use the subscript to indicate the URLLC link with and the eMBB link with . The subscript identifies OMA and NOMA, respectively.
Iia SingleCell Channel Model with OMA and NOMA
In a downlink scenario, we consider a single user that is associated with a single base station (BS). For both OMA and NOMA, the transmissions of the BS at a given time occupy up to the frequency bandwidth normalized to one, which is divided into the number of miniblocks. Each miniblock is assumed to be within the frequencytime channel coherence intervals. The channel coefficients are thus constants within each miniblock, and fade independently across different miniblocks over frequency and time. The transmit power of the BS is equally divided for each miniblock, normalized to one.
IiA1 SingleCell OMA
A set of miniblocks are allocated to , with . Each set corresponds to a fraction , with . The transmit power allocations to and are set as the maximum transmit power per miniblock. Denoting as the transmit power allocation fraction to miniblock , this corresponds to the allocations that equal .
The user’s received signaltonoise ratio () is determined by smallscale and largescale fading gains. For a given userBS association distance , the largescale fading gains of and are identically given as with the path loss exponent . For miniblock , the smallscale fading gain
is an exponential random variable with unit mean, which is independent and identically distributed (i.i.d.) across different miniblocks. The user’s received
of through miniblock is then expressed as(1) 
where is the noise spectral density of a single miniblock.
IiA2 SingleCell NOMA
The entire bandwidth is utilized for both links in NOMA, i.e., the miniblock allocation fractions equal . This is enabled by transmitting the superposition of the signals intended for and , with their different transmit power allocations, and then by decoding the signals with SIC at reception [3GPPMUST:2015]. The transmit power allocated to has a fraction of the maximum transmit power per miniblock, with .
At reception, unless otherwise noted, we consider is decoded prior to . This SIC order implicitly captures the lowlatency guarantee of , as addressed in [Petar5G:18] for an uplink scenario. Furthermore, it improves the overall NOMA system performance due to the reliability diversity of and , to be elaborated in Sect. LABEL:Sect:OptNOMA.
With the said SIC order, the signal intended for is first decoded, while treating the signal for as noise, i.e., multimodal selfinterference. The decoded signal is then removed by applying SIC, and the remaining signal for is finally decoded without selfinterference. The user’s received for through miniblock is thereby obtained as
(2)  
(3) 
Note that all the fading gains of and are identically since their channels are identical.
IiB Channel Model under a Stochastic Geometric Network
By using stochastic geometry, the aforementioned singlecell operation of OMA and NOMA is extended to a largescale multicell scenario as follows. The BSs under study are deployed in a twodimensional Euclidean plane, according to a stationary Poisson point process (PPP) with density , where the coordinates of a BS belongs to . Following the singlecell operation, each BS serves a single user through its and .
The locations of users follow an arbitrary stationary point process. Each user associates with the nearest BS, and downloads the visuohaptic VR traffic through the and of the BS. Following [Andrews:2011bg], we focus our analysis on a typical user that is located at the origin and associated with the nearest BS located at position of the plane. This typical user captures the spatiallyaveraged performance, thanks to Slyvnyak’s theorem [HaenggiSG] and the stationarity of .
In the previous singlecell scenario, interference occurs only from the multimodal selfinterference under NOMA, as shown in (2). In addition to such intracell selfinterference, extension to the stochastic geometric network model induces intercell interference. As done in [Andrews:2011bg, JHParkTWC:15, UR2Cspaswin:17], intercell interference is treated as noise, and is assumed to be large such that the maximum noise power is negligible. In this interferencelimited regime, channel quality is measured not by but by signaltointerference ratio (), as described next.
The intercell interference is measured by the typical user, and comes from the set of the BSs that are not associated with the typical user. We consider every BS always utilizes the entire bandwidth and the maximum transmit power. The average intercell interference per miniblock is thus identically given under both OMA and NOMA. The instantaneous intercell interference varies due to smallscale fading. For each miniblock, any interfering link’s smallscale fading is independent of the smallscale fading of the typical user’s desired and .
Under OMA, the typical user’s received of through miniblock is thereby given as
(4) 
where ’s are exponential random variables with unit mean, which are independent of and are i.i.d. across different interfering BSs. Likewise, under NOMA, the typical user’s received of through miniblock is expressed as
(5)  
(6) 
It is noted that all the s under NOMA and OMA are identically distributed across different miniblocks. For the typical user’s and , the largescale fading gains are identical. Their smallscale fading gains are independent under OMA, but are fullycorrelated under NOMA.
IiC Average Rate with Decoding Success Guarantee
In a largescale downlink cellular system with OMA and NOMA, we derive the typical user’s average rate that guarantees a target decoding success probability. Decoding becomes successful when the instantaneous downlink rate exceeds the transmitted coding rate.
To facilitate tractable analysis, we consider that the instantaneous channel information is not available at each BS. With the channel information at a BS, one can improve the average rate by adjusting the transmit power [Petar5G:18] and/or the coding rate [Andrews:2011bg, JHParkTWC:15]. In addition, we assume separate coding for each miniblock, which may loose frequency diversity gain compared to the coding across multiple miniblocks [Petar5G:18, TseBook:FundamaentalsWC:2005].
With these assumptions and the s that are identically distributed across miniblocks, average rate is determined by the decoding success probability for any single miniblock. Therefore, we drop the superscript in and the smallscale fading terms, and derive the average rate in the sequel.
IiC1 Oma
The typical user can decode the signal from with the decoding success probability that equals
(7)  
(8) 
where is the coding rate per miniblock, which is hereafter rephrased as a target threshold .
For the given target decoding success probability of , the average rate of is obtained by using outage capacity [TseOC:07] as
(9)  
(10) 
where the optimal target threshold satisfies , and thus equals .
Note that even when the coding block length of is short, the average rate expression in (10) still holds, since the finiteblock length rate under fading channels converges to the outage capacity [DurisiPolyanski:14].
IiC2 Noma
With the SIC order that decodes prior to , the typical user’s decoding success probabilities and of and are given as
(11)  
(12) 
Following [JindalSIC:09], our SIC do not allow to decode the signal after the decoding failure of the signal. With a different SIC architecture that allows such a decoding attempt, (12) is regarded as the lower bound, as done in [Petar5G:18].
For the given target decoding success probability of , the average rate of is given as
(13)  
(14) 
where the optimal target threshold equals . Similarly, for the given target decoding success probability of , the average rate of is given as
(15)  
(16) 
where .
IiD VisuoHaptic Perceptual Resolution
The resolution of human perceptions is often measured by using JND in psychophysics. In a psychophysical experiment, the JND is calculated as the minimum stimulus variation that can be detectable during % of the trials [Ernst:2002aa]. For a visuohaptic perception, its integrated JND is obtained by combining the JNDs of visual and haptic perceptions.
To elaborate, when individual haptic and visual perceptions have the perceived noise variances
and , a human brain combines these perceptions, yielding an integrated noise variance that satisfies . This relationship was first discovered in [Ernst:2002aa] by measuring the corresponding JNDs that are proportional to the perceived noise variances. The said relationship is thus read as , where denotes the JND measured when using both visuohapric perceptions, while and identify the JNDs of the individual haptic and visual perceptions, respectively.For individual visual perceptions, it has been reported by another experiment [Shi:10] that the PER is proportional to its sole JND due to the resulting visual frame loss. Similarly, for individual haptic perceptions, it has been observed in [ShiHirche:16] that the PER is proportional to the elapsed time to complete a given experimental task, which increases with the corresponding JND due to the coarse perceptions. Based on such experimental evidence, we can write that , where represents the PER on .
Accordingly, the JND of visuohaptic perceptions is obtained from the following equation
(17) 
In the following subsection, we adjust the target decoding success probabilities and of and , so as to guarantee a target visuohaptic JND , i.e., .
IiE URLLCeMBB Multiplexing Problem Formulation
In a downlink cellular system serving visuohaptic VR traffic, haptic and visual perceptions are supported through and , respectively. Each link pursues different service objectives as follows. The URLLC aims at:

Ensuring a target decoding success probability ; and

Ensuring a target average rate .
In contrast, the eMBB aims at:

Maximizing the average rate ; while

Ensuring a target decoding success probability , with .
In (iv), follows from an experimental evidence that the quality of visual perceptions dramatically drops when PER exceeds a certain limit, e.g., % PER that equals [Shi:10].
In addition to these individual service objectives, with and , their aggregate JND should guarantee a target visuohaptic JND . The said service objectives and requirements of and are described in the following problem formulation.
(18a)  
(18b)  
(18c)  
(18d) 
The objective functions and in the constraint (18b) are obtained from (10) for OMA and from (14) for NOMA. In the constraint (18c), is provided in (17). Without loss of generality, we hereafter consider a sufficiently large number of miniblocks so that the miniblock allocation fraction under OMA is treated as a continuous value.
Iii Optimal Multiplexing of VisuoHaptic VR Traffic under OMA and NOMA
In this section, we optimize the multiplexing of and that support visuohaptic VR traffic. With P1, for OMA, we optimize the miniblock allocation from the unit frequency block to each link. For NOMA, on the other hand, we optimize the power allocation from the unit transmit power.
Iiia Optimal OMA
We aim at optimizing the miniblock allocation . To this end, for given and , we derive the average rate with . This requires taking the inverse function of in (8).
The typical user’s is commonly referred to as coverage probability, and its closedform expression can be derived by using stochastic geometry [Andrews:2011bg, Haenggi:ISIT14]. Namely, is given as
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