Joint Scheduling of URLLC and eMBB Traffic in5G Wireless Networks

Arjun Anand*, Gustavo de Veciana*, and Sanjay Shakkottai*

*Department of Electrical and Computer Engineering, The University of Texas at Austin

Abstract—Emerging 5G systems will need to efficiently supportboth enhanced mobile broadband traffic (eMBB) and ultra-low-latency communications (URLLC) traffic. In these systems, timeis divided into slots which are further sub-divided into minislots.From a scheduling perspective, eMBB resource allocations occurat slot boundaries, whereas to reduce latency URLLC trafficis pre-emptively overlapped at the minislot timescale, resultingin selective superposition/puncturing of eMBB allocations. Thisapproach enables minimal URLLC latency at a potential rateloss to eMBB traffic.

We study joint eMBB and URLLC schedulers for such systems,with the dual objectives of maximizing utility for eMBB trafficwhile immediately satisfying URLLC demands. For a linear rateloss model (loss to eMBB is linear in the amount of URLLCsuperposition/puncturing), we derive an optimal joint scheduler.Somewhat counter-intuitively, our results show that our dualobjectives can be met by an iterative gradient scheduler foreMBB traffic that anticipates the expected loss from URLLCtraffic, along with an URLLC demand scheduler that is obliviousto eMBB channel states, utility functions and allocation decisionsof the eMBB scheduler. Next we consider a more general classof (convex/threshold) loss models and study optimal online jointeMBB/URLLC schedulers within the broad class of channel statedependent but minislot-homogeneous policies. A key observationis that unlike the linear rate loss model, for the convex andthreshold rate loss models, optimal eMBB and URLLC schedul-ing decisions do not de-couple and joint optimization is necessaryto satisfy the dual objectives. We validate the characteristics andbenefits of our schedulers via simulation.

Index Terms—wireless scheduling, URLLC traffic, 5G systems

I. INTRODUCTION

An important requirement for 5G wireless systems is itsability to efficiently support both broadband and ultra reliablelow-latency communications. On one hand enhanced MobileBroadband (eMBB) might require gigabit per second datarates (based on a bandwidth of several 100 MHz) and amoderate latency (a few milliseconds). On the other hand,Ultra Reliable Low Latency Communication (URLLC) trafficrequires extremely low delays (0.25-0.3 msec/packet) withvery high reliability (99.999%) [1]. To satisfy these heteroge-nous requirements, the 3GPP standards body has proposed aninnovative superposition/puncturing framework for multiplex-ing URLLC and eMBB traffic in 5G cellular systems1.

The proposed scheduling framework has the followingstructure [1]. As with current cellular systems, time is divided

1An earlier version of this work appears in the Proceedings of IEEEInfocom 2018, Honolulu, HI, [2].

time

Band

width(frequ

ency)

slott slot(t+1)

minislot

eMBB user1

eMBB user2

OverlappedURLLCtraffic

Fig. 1. Illustration of superposition/puncturing approach for multiplexingeMBB and URLLC: Time is divided into slots, and further subdivided intominislots. eMBB traffic is scheduled at the beginning of slots (sharing fre-quency across two eMBB users), whereas URLLC traffic can be dynamicallyoverlapped (superpose/puncture) at any minislot.

into slots, with a proposed one millisecond (msec) slot dura-tion. Within each slot, eMBB traffic can share the bandwidthover the time-frequency plane (see Figure 1). The sharingmechanism can be opportunistic (based on the channel statesof various users); however, the eMBB shares are decided bythe beginning, and fixed for the duration of a slot2. Furtherthe new framework also allows aggregation of eMBB slotswhere transmissions to an eMBB user over consecutive slotsare coded together to achieve better coding gains resultingfrom long codewords while reducing overheads due to controlsignals. This results in better spectral efficiency as comparedto the OFDMA frame structure of LTE [3].

URLLC downlink packets may arrive during an ongoingeMBB transmission; if tight latency constraints are to besatisfied, they cannot be queued until the next slot. Insteadeach eMBB slot is divided into minislots, each of which hasa 0.125 msec duration3. Thus upon arrival URLLC packetscan be immediately scheduled in the next minislot on topof the ongoing eMBB transmissions. If the Base Station(BS) chooses non-zero transmission powers for both eMBBand overlapping URLLC traffic, then this is referred to as

2The sharing granularity among various eMBB users is at the level ofResource Blocks (RB), which are small time-frequency rectangles within aslot. In LTE today, these are (1 msec × 180 KHz), and could be smaller for5G systems.

3In 3GPP, the formal term for a ‘slot’ is eMBB TTI, and a ‘minislot’ is aURLLC TTI, where TTI expands to Transmit Time Interval.

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superposition. If eMBB transmissions are allocated zero powerwhen URLLC traffic is overlapped, then it is referred to aspuncturing of eMBB transmissions. To achieve high reliabilityURLLC transmissions are by design protected through codingand HARQ if necessary. At the end of an eMBB slot, theBS can signal eMBB users the locations, if any, of URLLCsuperposition/puncturing. eMBB users can then use this infor-mation to decode transmissions, with some possible loss ofrate depending on the amount of URLLC overlap. See [1], [4]for additional details.

A key problem in this setting is the joint scheduling ofeMBB and URLLC traffic over two time-scales. At the slotboundary, resources are allocated to eMBB users (with pos-sible aggregation of slots) based on their channel states andutilities, in effect, allocating long term rates to optimize high-level goals (e.g. utility optimization). Meanwhile, at each min-islot boundary, the (stochastic) URLLC demands are placedonto previously scheduled and ongoing eMBB transmissions.Decisions on the placement of such overlaps across scheduledeMBB user(s) will impact the rates they will see on that slot.Thus we have a coupled problem of jointly optimizing thescheduling of eMBB users on slots with the placement ofURLLC demands across minislots.

A. Main Contributions

This paper is, to our knowledge, the first to formalize andsolve the joint eMBB/URLLC scheduling problem describedabove. We consider various models for the eMBB rate lossassociated with URLLC superposition/puncturing, for whichwe characterize the associated feasible throughput regions andpropose online joint scheduling algorithms as detailed below.Linear Model: When the rate loss to eMBB is directly pro-portional to the fraction of superposed/punctured minislots, weshow that the joint optimal scheduler has a nice decomposition.Despite having non-linear utility functions and time-varyingchannel states, the stochastic URLLC traffic can be uniform-randomly placed in each minislot, while the eMBB schedulercan be scheduled via a greedy iterative gradient algorithm thatonly accounts for the expected rate loss due to the URLLCtraffic.Convex Model: For more general settings where the rateloss can be modeled by a convex function, the solution doesnot have the decomposition property as in the linear modeland hence, the finding the optimal solution is challenging.Therefore, we restrict to a simpler class of joint schedulingpolicies called as minislot-homogeneous joint scheduling poli-cies where the URLLC placement policy does not changeacross the minislots in an eMBB slot. In this setting, we char-acterize the capacity region and derive concavity conditionsunder which we can derive the effective rate seen by eMBBusers (post-puncturing by URLLC traffic). We then developa stochastic approximation algorithm which jointly scheduleseMBB and URLLC traffic, and show that it asymptoticallymaximizes the utility for eMBB users while satisfying URLLCdemands. We also show that for convex functions which arehomogeneous, minislot-homogeneous joint scheduling poli-

cies are optimal within the larger class of causal and non-anticipative joint scheduling policies. Further for the convexloss model, we show that it is better to schedule eMBB usersto share bandwidth (i.e. slice across frequency, see also Fig. 4),and let each user occupy the entire slot duration to mitigaterate loss due to URLLC puncturing.Threshold Model: Finally we consider a loss model, whereeMBB traffic is unaffected by puncturing until a threshold isreached; beyond this threshold it suffers complete throughputloss (a 0-1 rate loss model). We consider two broad classesof minislot homogeneous policies, where the URLLC trafficis placed in minislots in proportion to the eMBB resourceallocations (Rate Proportional (RP)) or eMBB loss thresholds(Threshold Proportional (TP)). We motivate these policies (e.g.TP minimizes the probability of any eMBB loss in an eMBBslot) and derive the associated throughput regions. Finally,we utilize the additional structure underlying the RP and TPPlacement policies along with the shape of the threshold lossfunction to derive fast gradient algorithms that converge andprovably maximize utility.

B. Related Work

Resource allocation, utility maximization and opportunisticscheduling for downlink wireless systems have been intenselystudied in the last two decades, and have had a major impacton cellular standards. We refer to [5], [6] for a survey of thekey results. In this paper, we focus on joint scheduling ofURLLC and eMBB traffic. From an application point of view,there have been several studies arguing for the need to supportURLLC services (e.g. for industrial automation) [7], [8], [9].

With demand of both broadband and low-latency servicesgrowing, there has been rapid developments in the 5G stan-dardization efforts in 3GPP. Of key relevance to this paper, the3GPP RAN WG1 has focused on standardizing slot structurefor eMBB and URLLC, and have been evaluating signalingand control channels to support superposition and puncturingin recent meetings [1], [4]. We specifically refer the reader toSections 8.1.1.3.4 – 8.1.1.3.6 in [4] for current proposals.

Beyond standards, recent work has focused on system leveldesign for such systems (overheads, packet sizes, controlchannel structure, etc.) [3], [10], [11]. Of particular note,[10] argues (based on system level simulation and queuingmodels) that statically partitioning bandwidth between eMBBand URLLC is very inefficient. There have also been severalstudies focusing on physical layer aspects of URLLC (codingand modulation, fading, link budget) [12], [13].

Efficient sharing of radio resources between eMBB andURLLC traffic has been discussed in literature, see [14], [15],[16]. In [14], the authors have considered joint optimization ofresource allocation for eMBB and URLLC traffic. However,they do not use puncturing/superposition mechanisms to shareresources. Some works ([15], [16]) use information theoreticresults to obtain expressions for the average eMBB ratesunder URLLC puncturing for various decoding schemes foruplink eMBB traffic punctured/superposed by URLLC users.However, they do not consider the design of joint scheduling

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algorithms for eMBB and URLLC traffic. To the best ofour knowledge, our paper is the first to explore the resourceallocation issues for joint scheduling of URLLC and eMBBtraffic using puncturing/superposition based mechanisms.

II. SYSTEM MODEL

Traffic model: We consider a wireless system supportinga fixed set U of backlogged eMBB users and a stationaryprocess of URLLC demands. eMBB scheduling decisions aremade across slots while URLLC demands arrive and areimmediately scheduled in the next minislot. In this section weshall consider the case where eMBB all users receive resourcesfor slots without using slot aggregation even though moreflexible resource allocations which can possibly include slotaggregation and splitting are proposed in 5G standards [3]. Weshall justify this choice in Sec. IV-E. Each eMBB slot has anassociated set of minislots where the set M = {1, . . . |M|}denotes their indices. URLLC demands across minislots aremodeled as an independent and identically distributed (i.i.d.)random process. We let the random variables (D(m),m ∈M)denote the URLLC demands per minislot for a typical eMBBslot and let D be a random variable whose distribution isthat of the aggregate URLLC demand per eMBB slot, i.e.,D ∼

∑m∈MD(m) with, cumulative distribution function

FD(·) and mean E[D] = ρ. We assume demands have beennormalized so the maximum URLLC demand per minislotis f and the maximum aggregate demands per eMBB slotis f × |M| = 1 i.e., all the frequency-time resources areoccupied. URLLC demands per minislot exceeding the systemcapacity are blocked by URLLC scheduler thus D ≤ 1 almostsurely. The system is engineered so that blocked URLLCtraffic on a minislot is a rare event, i.e., satisfies the desiredreliability on such traffic.

Wireless channel variations: The wireless system experi-ences channel variations each eMBB slot which are modeledas an i.i.d. random process over a set of channel statesS = {1, . . . , |S|}. Let S be a random variable modelingthe distribution over the states in a typical eMBB slot withprobability mass function pS(s) = P (S = s) for s ∈ S. Foreach channel state s eMBB user u has a known peak ratersu. The wireless system can choose what proportions of thefrequency-time resources to allocate to each eMBB user oneach minislot for each channel state. This is modeled by amatrix φ ∈ Σ where

Σ :={φ ∈ R|U|×|M|×|S|+ |∑

u∈Uφsu,m = f, ∀m ∈M, s ∈ S

}(1)

and where the element φsu,m represents the fraction of re-sources allocated to user u in mini slot m in channel states. We also let φsu =

∑m∈M φsu,m, i.e., the total resources

allocated to user u in an eMBB slot in channel state s.Now assuming no superposition/puncturing if the system is inchannel state s and the eMBB scheduler chooses an allocation

φ the rate ru allocated to user u would be given by ru = φsursu.

The scheduler is assumed to know the channel state andcan thus opportunistically exploit such variations in allocatingresources to eMBB users. Note that for simplicity, we adopta flat-fading model, namely, the rate achieved by an user isdirectly proportional to the fraction of bandwidth allocated toit (the scaling factor is the peak rate of the user for the currentchannel state).

Class of joint eMBB/URLLC schedulers: We considera class of stationary joint eMBB/URLLC schedulers denotedby Π satisfying the following properties. A scheduling policycombines a possibly state dependent eMBB resource allo-cation matrix φ per slot with a URLLC demand placementstrategy across minislots. The placement strategy may impactthe eMBB users’ rates since it affects the URLLC superpo-sition/puncturing loads they will experience. As mentionedearlier in discussing the traffic model, in order to meet lowlatency requirements URLLC traffic demands are scheduledimmediately upon arrival or blocked. The scheduler is assumedto be causal so it only knows the current (and past) channelstates and peak rates rsu for all u ∈ U and s ∈ S but doesnot know the realization of future channels or URLLC trafficdemands. In making superposition/puncturing decisions acrossminislots, the scheduler can use knowledge of the previousplacement decisions that were made. In addition the scheduleris assumed to know (or able measure over time) the channelstate distribution across eMBB slots and URLLC demanddistributions per minislot i.e., that of D(m), and per eMBBslot, i.e., D, and thus in particular knows ρ = E[D].

In summary a joint scheduling policy π ∈ Π is thuscharacterized by the following:• an eMBB resource allocation φπ ∈ Σ where φπ,su,m

denotes the fraction of frequency-time slot resourcesallocated to eMBB user u on minislot m when the systemis in state s.

• the distributions of URLLC loads across eMBB re-sources induced by its URLLC placement strategy, de-noted by random variables Lπ = (Lπ,su,m|u ∈ U ,m ∈M, s ∈ S) where Lπ,su,m denotes the URLLC loadsuperposed/puncturing the resource allocation of user uon minislot m when the channel is in state s.

The distributions of Lπ,su,m and their associated means lπ,s

u,m

depend on the joint scheduling policy π, but for all states,users and minislots satisfy

Lπ,su,m ≤ φπ,su,m almost surely.

In the sequel we let Lπ,su =∑m∈M Lπ,su,m, i.e., the aggregate

URLLC traffic superposed/puncturing user u in channel states, and denote its mean by l

π,s

u and note that

Lπ,su ≤ φπ,su almost surely.

We also let Lπ,s :=∑u∈U L

π,su denote the aggregate induced

load and note that any policy π and for any state s we havethat

ρ = E[D] = E[Lπ,s] = E[∑u∈U

Lπ,su ] =∑u∈U

lπ,s

u .

3

h (x)us

x

1

10

success (if any) with full overlap

linear model

threshold model

convex model

Fig. 2. The illustration exhibits the rate loss function for the various modelsconsidered in this paper, linear, convex and threshold.

Modeling superposition/puncturing and eMBB capacityregions: Under a joint scheduling policy π we model the rateachieved by an eMBB user u in channel state s by a randomvariable

Rπ,su = fsu(φπ,su , Lπ,su ), (2)

where the rate allocation function fsu(·, ·) models the impactof URLLC superposition/puncturing – one would expect it tobe increasing in the first argument (the allocated resources)and decreasing in the second argument (the amount superpo-sition/puncturing by URLLC traffic). Under our system modelwe have that

Rπ,su ≤ fsu(φπ,su , 0) = φπ,su rsu almost surely,

with equality if there is no superposition/puncturing, i.e., whenlsu = 0. Let rπ,su = E[Rπ,su ] denote the mean rates achieved byuser u in state s under the URLLC superposition/puncturingdistribution induced by scheduling policy π.Models for Throughput Loss: In the sequel we shall considerspecific forms of superposition/puncturing loss models: (i)linear, (ii) convex, and (iii) threshold models.

We rewrite the rate allocation function in (2) as the differ-ence between the peak throughput and the loss due to URLLCtraffic, and consider functions that can be decomposed as:

fsu(φsu, lsu) = rsuφ

su

(1− hsu

(Lπ,suφsu

)),

where hsu : [0, 1] → [0, 1] is the rate loss function andcaptures the relative rate loss due to URLLC overlap oneMBB allocations. The puncturing models we study now mapdirectly to structural assumptions on the rate loss functionhsu(·); namely it is a non-decreasing function, and is one oflinear, convex, or threshold as shown in Figure 2.Linear Model: Under the linear model, the expected rate foruser u in channel state s for policy π is given by

rπ,su = E[fsu(φπ,su , Lπ,su )] = rsu(φπ,su − lπ,s

u ),

i.e., hsu(x) = x, and the resulting rate to eMBB users is a linearfunction of both the allocated resources and mean inducedURLLC loads. This model is motivated by basic results forthe channel capacity of AWGN channel with erasures, see [17]for more details. Our system in a given network state can

be approximated as an AWGN channel with erasures, whenthe slot sizes are long enough so that the physical layererror control coding of eMBB users use long code-words.Further, there is a dedicated control channel through whichthe scheduler can signal to the eMBB receiver indicating thepositions of URLLC overlap. Indeed such a control channelhas been proposed in the 3GPP standards [1]. Note thatunder this model the rate achieved by a given user dependson the aggregate superposition/puncturing it experiences, i.e.,does not depend on which minislots and frequency bands itoccurs. We discuss scheduling policies for linear loss modelsin Section III.Convex Model: In the convex model, the rate loss functionhsu(·) is convex (see Figure 2), and the resulting rate for eMBBuser u in channel state s under policy π is given by

rπ,su = E[fsu(φπ,su , Lπ,su )] = rsuφπ,su

(1− E

[hsu

(Lπ,suφπ,su

)]).

This covers a broad class of models, and is discussed inSection IV.Threshold Model: Finally the threshold model is designed tocapture a simplified packet transmission and decoding processin an eMBB receiver. The data is either received perfectly or itis lost depending on the amount of superposition/puncturing.With slight abuse of notation we shall let hsu also depend onboth the relative URLLC load and the eMBB user allocation,i.e., hsu(x) = 1(x ≥ tsu(φsu)) where the threshold in turn isan increasing function tsu(·) satisfying x ≥ tsu(x) ≥ 0. Suchthresholds might reflect various engineering choices wherecodes are adapted when users are allocated more resources,so as to be more robust to interference/URLLC superposi-tion/puncturing. The resulting rate for eMBB user u in channelstate s and policy π is then given by

rπ,su = rsuφπ,su P (Lπ,su ≤ φπ,su tsu(φπ,su )).

While such a sharp falloff is somewhat extreme, it is never-theless useful for modeling short codes that are designed totolerate a limited amount of interference. In practice one mightexpect a smoother fall off, perhaps more akin to the convexmodel, e.g., when hybrid ARQ (HARQ) is used. We discusspolices under the threshold based model in Section V.Capacity set for eMBB traffic: We define the capacity setC ⊂ R|U|+ for eMBB traffic as the set of long term ratesachievable under policies in Π. Let cπ = (cπu|u ∈ U) where

cπu =∑s∈S

rπ,su pS(s).

Then the capacity is given by

C = {c ∈ R|U|+ | ∃ π ∈ Π such that c ≤ cπ}.

Note that this capacity region depends on the schedulingpolicies under consideration as well as the distributions of thechannel states and URLLC demands.Scheduling objective: URLLC priority and eMBB utilitymaximization: As mentioned earlier, URLLC traffic is im-mediately scheduled upon arrival, in the next minislot, i.e,

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no queuing is allowed. Thus if demands exceed the systemcapacity on a given minislot traffic would be lost. However,we assume that the system has been engineered so thatsuch URLLC overloads are extremely rare, and thus URLLCtraffic can meet extremely low latency requirements with highreliability4. For eMBB traffic we adopt a utility maximizationframework wherein each eMBB user u has an associated utilityfunction Uu(·) which is a strictly concave, continuous anddifferentiable of the average rate cπu experienced by the user.Our aim is to characterize optimal rate allocations associatedwith the utility maximization problem:

maxc{∑u∈U

Uu (cu) | c ∈ C}, (3)

and determine a scheduling policy π that will realize suchallocations.

III. LINEAR MODEL FOR SUPERPOSITION/PUNCTURING

In any state s, the optimal joint eMBB/URLLC schedulermay either 1) protect the user with the lower channel rateby placing less URLLC traffic into its frequency resources toensure fairness or 2) opportunistically place URLLC trafficso that the user with a better channel gets a higher rateto improve the overall system throughput. The solution forany state is complex function of network states and theirdistribution and user utility functions and in general, eMBBscheduling and URLLC puncturing may be dependent. In thissection, we show a surprising result – despite having non-linear utility functions, if the loss functions are linear and theeMBB scheduler is intelligent (i.e., takes into the degradationof rates due to puncturing), then the URLLC scheduler can beoblivious to the channel states, utility functions and the actualrate allocations of the eMBB scheduler.

A. Characterization of capacity region

Let us consider the capacity region for a wireless systembased on linear superposition/puncturing model under a re-stricted class of policies ΠLR that combine feasible eMBBallocations φ ∈ Σ with random placement of URLLC de-mands uniformly over the bandwidth across minislots. Notethat the notation LR stands for linear loss model (L) withrandom (R) placement of URLLC traffic. For any π ∈ ΠLR

with eMBB allocation φπ the mean induced loads under suchrandomization for each state s ∈ S and minislot m ∈ Mwill satisfy l

π,s

u,m = ρφπ,su,m. Indeed randomization clearlyleads to an induced loads that are proportional to the eMBBallocations on a per mini-slot basis, but also per eMBB slot,i.e., l

π,s

u = ρφπ,su . Thus for our linear loss model we have that

rπ,su = rsu(φπ,su − lπ,s

u ) = rsuφπ,su (1− ρ).

Hence the overall user rates achieved under such a policy aregiven by cπ = (cπu|u ∈ U) where

cπu =∑s∈S

rsuφπ,su (1− ρ)pS(s).

4Note that since we allow URLLC traffic in the entire system bandwidth,such overload events are very rare.

The capacity region associated with policies that use URLLCuniformly randomized placement is thus given by

CLR = {c ∈ R|U|+ | ∃π ∈ ΠLR s.t. c ≤ cπ}= {c ∈ R|U|+ | ∃φ ∈ Σ s.t. c ≤ cφ},

where we have abused notation by using cφ to represent thethroughput achieved under policy π that uses eMBB resourceallocation φ and uniformly randomized URLLC demandplacement. Finally note that for any fixed ρ ∈ (0, 1), CLR is aclosed and bounded convex region. This is because an affinemap of a convex region remains convex; hence multiplying theconstraints on the capacity region defined by φ by a constant(1− ρ) preserves convexity of the rate region.

Theorem 1. For a wireless system under the linear superpo-sition/puncturing loss model we have that C = CLR.

The proof is deferred to the Appendix A. In other wordsthe throughput cπ ∈ C achieved by any feasible policy π ∈ Πcan also be achieved by policy π′, with a possibly differenteMBB resource allocation policy than π but utilizing uniformrandom placement of URLLC demands across mini-slots.

B. Utility maximizing joint scheduling

Given the result in Theorem 1 we now restate the utilitymaximization problem as optimizing solely over joint schedul-ing policies that use URLLC random placement policies, asfollows:

maxφ∈Σ

∑u∈U

Uu(cφu ),

s.t. cφu =∑s∈S

rsuφsu(1− ρ)pS(s), ∀u ∈ U .

The above optimization problem has a strictly concave costfunction and convex constraints. Thus, at face-value, it appearsthat we can apply the gradient scheduler introduced in [18],which is an online algorithm designed to converge to thesolution of similar optimization problem. This observation isapproximately correct, but subject to two modifications.

First, the setting in [18] has deterministic rates in eachchannel state. However, in our case, in each channel state,the rates are stochastic due to puncturing by URLLC traffic(this results in the (1 − ρ) correction). This can be easilyaddressed by modifying the setting in [18]; the finite state andi.i.d. nature of puncturing implies that the proofs in [18] holdwith minor modifications; we skip the details.

The second issue is somewhat more nuanced. In currentwireless systems (e.g. LTE) and proposals for 5G systems, aslot is partitioned into a collection of Resource Blocks (RB),where each RB is a time-frequency rectangle (1 msec × 180KHz in LTE). Importantly, these RBs can be individuallyallocated to different eMBB users. If we now apply thegradient scheduler in [18] to our setting, the result will be thatall RBs in a slot will be allocated to the same user. While this isno-doubt asymptotically optimal, it seems intuitive that sharingRBs across users even within a slot will lead to better short-term performance. Indeed this intuition has been explored

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in the context of iterative MaxWeight algorithms to provideformal guarantees, see [19], [20]. The high level idea is thateven within a slot, RB allocations are done iteratively, wherefuture RB allocations need to account for prior rate allocationseven within the same slot. This is formalized below, where wedescribe our proposed joint eMBB-URLLC scheduler.

The URLLC scheduler: As explained in the previoussection, the URLLC scheduler places the URLLC trafficuniformly at random in each minislot.

The eMBB scheduler: Let there be B resource blocksavailable for allocation every eMBB slot, indexed by1, 2, . . . , B. Let Ru(t−1) be the random variable denoting theaverage rate received by eMBB user up to eMBB slot t − 1.Let ru(t−1) be a realization of Ru(t−1). In any eMBB slott we schedule an user u(b) in RB b such that

u(b) ∈ argmax{rsuU

′

u (rεu (b− 1, t)) , u = 1, 2, . . . ,U}, (4)

where rεu (b− 1, t) is an estimate of the average rate receivedby eMBB user u till slot t which is iteratively updated asfollows:

rεu (b, t) =

ru(t− 1), b = 0,

(1− ε) rεu (b− 1, t)

+ε(rsu

1B (1− ρ)1 (i = u(b))

), b 6= 0.

(5)

In the above equation, ε is a small positive value. At the endof eMBB slot t, the eMBB scheduler receives feedback fromthe eMBB receivers indicating the actual rates received by theeMBB users due to allocations. We denote the rate receivedeMBB user u in slot by the random variable Ru(t) and itsrealization by ru(t). We finally update ru(t) as follows:

ru(t) = (1− ε) ru(t− 1) + εru(t). (6)

This scheduler and update equations are analogous to thegradient algorithm [18] (see also iterative algorithms in [19],[20]). The optimality proof of this algorithm follows (withminor modifications) from the analysis in [18]; we skip thedetails.Remarks: (i) A natural decomposition of the jointeMBB+URLLC scheduling is now apparent. On one hand,the eMBB scheduler maximizes utilities based on the expectedchannel rates stemming from uniform random puncturing ofminislots (accounted for through the (1 − ρ) multiplicativefactor), and does so using the iterative gradient scheduler. TheURLLC scheduler, on the other-hand, is completely agnosticto either the channel state or the actual eMBB allocations andsimply punctures minislots based on the current instantaneousdemand.

(ii) The fact that the URLLC traffic placement is completelyagnostic to the channel state and eMBB utilities/allocationis surprising. Intuitively it seems plausible that one couldpuncture an eMBB user with a lower marginal utility withmore URLLC traffic, while protecting an eMBB user witha higher marginal utility and achieve a better sum utility.Further, it seems reasonable that eMBB users with a worsechannel state (and thus lower rate) could be loaded with

additional URLLC traffic. However, Theorem. 1 implies thatthere exists an optimal solution that is achieved by channel andutility oblivious and uniform random URLLC placement, thusproviding a very simple algorithm for URLLC scheduling.

(iii) We remark that the optimality of random puncturingfor linear loss models depends critically on the use of an op-portunistic scheduler for eMBB traffic.To see this, consider asimple system with two symmetric eMBB users each with twopossible channel states. The associated channel rates are either{2, 4} packets/slot with equal probability, and independentacross users and time slots. Suppose that we use a static (non-opportunistic) scheduler, which equally splits channel accessbetween the users. It is easy to calculate that the rate to eachuser is then 1.5 packets/slot. Next suppose that the URLLCload is 50%, and that this traffic randomly punctures eMBBusers. Then from symmetry, it follows that the rate per eMBBuser is 0.75 packets/slot. In contrast, suppose that puncturingis opportunistic, where the user with the currently lower rateis punctured whenever possible (opportunistic puncturing ofthe currently worse eMBB user), a straightforward calculationshows that the rate to each eMBB user is 0.875 packets/slot,which is a strict improvement over random puncturing. At ahigh-level, this follows because opportunistic eMBB schedul-ing operates on the Pareto frontier of two-user capacity re-gion, and consequently there is no residual opportunistic tobe obtained by puncturing. However, with non-opportunisticscheduling, the system is not pushed to the boundary; thus,opportunistic puncturing can extract additional throughput foreMBB users.

IV. CONVEX MODEL – MINISLOT-HOMOGENOUSPOLICIES

In this section we shall consider joint scheduling for wire-less systems for convex superposition/puncturing loss models.This is a somewhat complex problem, whence we will focusour attention on a restricted, but still rich, class of schedul-ing policies which we refer to as minislot-homogeneouseMBB/URLLC schedulers. We identify a key concavity re-quirement in Assumption 2 (that is satisfied by convex lossfunctions) that enables a stochastic approximation approachfor utility maximizing scheduling.

A. Minislot-homogeneous eMBB/URLLC Scheduling policies

We shall define minislot-homogeneous eMBB/URLLCschedulers as follows. First, feasible eMBB allocations φ ∈ Σwill be restricted such that for any eMBB slot in channel states ∈ S allocations are minislot-homogeneous across minislotsin an eMBB slot, i.e., φsu,1 = φsu,m,∀m ∈ M and its overallallocation for the slot is given by φsu = |M|φsu,1. The set ofminislot-homogeneous eMBB allocations is thus given by

ΣH :={φ ∈ Σ | u ∈ U , φsu,m = φsu,1 ∀m ∈M,∀s ∈ S

}.

Second, URLLC demand placements per minislot are doneproportionally based on pre-specified weights, and theseweights are assumed to be time-homogeneous across minislots.In particular such policies are parametrized by a weight matrix

6

γ ∈ ΣH , where the induced load on user u under channel states and slot m is given by

Lsu,m =γsu,m∑

u′∈U γsu′,m

D(m) =γsu,1fD(m).

We shall call γsu,1 the URLLC placement factor for eMBB useru in state s. The eMBB and URLLC allocations are coupledtogether since it must be the case that for all u ∈ U Lsu,m ≤φsu,m = φsu,1 almost surely, i.e., one can not induce moresuperposition/puncturing load on a user than the resources ithas been allocated on that slot. So the following conditionmust be satisfied. For all m ∈M we have that

D(m) ≤ minu∈U

φsu,1γsu,1

f, almost surely.

Recall that f denotes the maximum URLLC load per minislotso D(m) ≤ f almost surely, thus if

φsu,1γsu,1

≥ 1 the abovecondition will always hold. Yet if φsu,1 ≥ γsu,1 for all u, thenwe have that φsu,1 = γsu,1, i.e., there is not flexibility to exploitcareful placement of URLLC demands. Hence, we introducethe following assumption:

Assumption 1. We say the system has a (1 − δ) URLLCsharing factor per minislot if D(m) ≤ f(1− δ) almost surelyfor all m ∈M, where δ ∈ (0, 1).

For any δ the above assumption implies that the peakURLLC demand in an eMBB slot can be at most 1 − δwhich is lower than maximum possible value of one. Suchan assumption is reasonable as we consider shared resourceswhich are engineered to meet the peak URLLC loads whilealso serving eMBB traffic. Under a (1 − δ) URLLC sharingfactor a minislot-homogeneous eMBB resource allocation φand URLLC allocation γ is will be feasible if for all s ∈ Swe have

(1− δ) ≤ minu∈U

φsu,1γsu,1

,

which is satisfied as long as (1− δ)γsu,1 ≤ φsu,1 for all u ∈ U .This motivates the following definition:

Definition 1. For a system with a (1 − δ) sharing factor,the feasible minislot-homogeneous eMBB/URLLC schedulingpolicies are parameterized by φ,γ ∈ ΣH such that (1−δ)γ ≤φ. We shall denote the set of such policies as follows:

ΠH,δ := {(φ,γ) | φ,γ ∈ ΣH and (1− δ)γ ≤ φ},

where ΠH,δ is a convex set.

B. Characterization of the throughput region

In this section we characterize the throughput regionsachievable under time-homogeneous scheduling.

Theorem 2. For a system with a (1 − δ) sharing factor andminislot-homogeneous scheduler π = (φπ,γπ) ∈ ΠH,δ theaverage induced throughput for user u ∈ U in channel states ∈ S is given by

rπ,su = E[fsu(φπ,su , γπ,su D)],

and the overall average user throughputs are given by cπ =(cπu | u ∈ U) where cπu =

∑s∈S r

π,su pS(s).

The proof is included in Appendix B. Based on the abovewe can define feasible throughput region constrained to thetime-homogeneous policies in ΠH,δ. First let us define

CH,δ = {c ∈ R|U|+ | ∃π ∈ ΠH,δ s.t. c ≤ cπ}

and let CH,δ denote the convex hull of CH,δ. Note that ratesin the convex hull are achievable through policies that dotime sharing/randomization amongst minislot-homogeneousscheduling policies in ΠH,δ.

Assumption 2. For all s ∈ S and u ∈ U the functions gsu(, )given by

gsu(φsu, γsu) = E[fsu(φsu, γ

suD)], (7)

are jointly concave on ΠH,δ.

Lemma 1. Assumption 2 is satisfied for systems where super-position/puncturing of each user is modelled via either a

1) Convex loss function or2) Threshold loss function with fixed relative thresholds,

i.e., tsu(φsu) = αsu for φ ∈ [0, 1] and the URLLC demanddistribution FD(·) is such that FD( 1

x ) is concave in x(satisfied by the truncated Pareto distribution).

The proof is included in Appendix C. With this conditionin place, we now describe the throughput region.

Theorem 3. Under Assumption 2 we have that CH,δ = CH,δ .

The proof is available in the Appendix D. The abovetheorem implies that we do not have to consider time-sharing/randomization amongst minislot-homogeneous jointscheduling policies. Thus, with minislot-homogeneous policiesand under the concavity of gπ,su (·, ·) from Assumption 2, theabove result sets up a convex optimization problem in (φ,γ),i..e, we have a concave cost function with convex constraints.Thus, by iteratively updating (φ,γ), we can develop an onlinescheduling algorithm that asymptotically maximizes eMBBusers’ utility. This is descried next.

C. Stochastic approximation based online algorithm

We first restate the utility maximization problem forminislot-homogeneous URLLC/eMBB scheduling policies:

maxφ,γ∈ΠU,δ

∑u∈U

Uu

(∑s∈S

pS(s)gsu (φsu, γsu)

). (8)

Observe that the objective function is concave because itconsists of a sum of compositions of non-decreasing concavefunctions (Uu(·)), and concave functions (gsu (·, ·)) in φ andγ (if Assumption 2 holds). Further, the constraint set isconvex. Therefore, the above problem fits in the frameworkof standard convex optimization problems. However, solvingthe above problem requires knowledge of all possible networkstates and their probability distribution, resulting in an offlineoptimization problem. In this section, we develop a stochastic

7

approximation based online algorithm to solve the aboveproblem.

Online algorithm: Let R(t − 1) :=(R1(t− 1), R2(t− 1), . . . , Ru(t− 1), . . . , R|U|(t− 1)

)be the random vector denoting the average rates received byeMBB users up to eMBB slot t−1 under our online algorithm.Let r(t − 1) denote a realization of R(t − 1). Let s be thenetwork state in slot t. Define vectors φs := (φsu, | u ∈ U)and γs := (γsu | u ∈ U). At the beginning of eMBB slot t, wecompute vectors

(φ(t), γ(t)

)as the solution to the following

optimization problem:

maxφs,γs

∑u∈U

U′

u (ru(t− 1)) gsu(φsu, γsu), (9)

s.t. φs ≥ (1− δ)γs, (10)∑u∈U

φsu = 1 and∑u∈U

γsu = 1, (11)

φs ∈ [0, 1]|U| and γs ∈ [0, 1]

|U|. (12)

This optimization problem is a convex optimization problemand can be solved numerically using standard convex optimiza-tion techniques. Using

(φ(t), γ(t)

), we schedule URLLC and

eMBB traffic as follows:The eMBB scheduler: For notational ease, we fluidize

the bandwidth. Specifically, we assume that the bandwidth ofa resource block is very small when compared to the totalbandwidth available. Hence, the bandwidth can be split intoarbitrary fractions and we allocate fraction φu(t) of the totalbandwidth to eMBB user u.

The URLLC Scheduler: We load different eMBB userswith URLLC traffic according to the vector γ(t).

At the end of eMBB slot t, the eMBB scheduler receivesfeedback from the eMBB receivers indicating the rates re-ceived by the eMBB users. Let us denote the rate receivedeMBB user u in the slot by the random variable Ru(t). Weupdate Ru(t) as follows:

Ru(t) = (1− εt)Ru(t− 1) + εtRu(t), (13)

where {εt | t = 1, 2, 3, . . .} is a sequence of positive numberswhich satisfy the following (standard) assumption:

Assumption 3. The averaging sequence {εt} satisfies:∞∑t=1

εt =∞ and∞∑t=1

ε2t <∞.

Finally, we state the main result of this section, which isthe optimality of the stochastic approximation based onlinealgorithm.

Theorem 4. Let r∗ be the optimal average rate vectorreceived by eMBB users under the solution to the offlineoptimization problem. Suppose that Assumptions 3 and 2 hold,then we have that:

limt→∞

R(t) = r∗ almost surely. (14)

The proof is available in the Appendix E.

D. Optimality of Minislot-Homogeneous Policies

In the previous section we restricted ourselves to minislot-homogeneous policies. In this section will justify this choice.Let us consider a generalization of minislot-homogeneouspolicies where the URLLC placement in each minislot candepend on the history of URLLC arrivals prior to that minislot.Such a policy will obviously perform better than minislot-homogeneous URLLC placement policies since in a minislot-homogeneous policy we decide the URLLC placement atthe beginning of an eMBB slot based on the expected lossdue to puncturing/superposition and do not adapt it based onthe realization of URLLC demands per minislot. However,finding an optimal scheduling policy under this generalizationcan be computationally expensive as compared to minislot-homogeneous policies which are attractive due to their simplic-ity. In this section we identify conditions under which minislot-homogeneous URLLC placement polices perform as well asthe general class of causal and minislot-dependent policies.These terms are defined below.

Definition 2. A scheduler is said to be causal if at the begin-ning of a mini-slot m the scheduler knows the realizations ofD(1), D(2), . . . , D(m− 1) and is unaware of the realizationsof D(m), D(m+ 1), . . . , D(|M|).

Definition 3. A scheduling policy is said to be minislot-dependent if the URLLC placement policy can vary with theminislot index m and previous URLLC demands in the eMBBslot.

The decision variables in a causal and minislot-dependentjoint scheduling policy π can be described as follows:

1) At the beginning of an eMBB slot, the scheduler choosesφπ,su , u ∈ U such that∑

u∈Uφπ,su = 1 and φπ,su ∈ [0, 1] ∀u ∈ U . (15)

2) In each mini-slot m, the total puncturing placed oneMBB user u is given by γπ,su,m

(d(1:m−1)

)Dm, where

γπ,su,m (·) characterizes the URLLC placement in min-islot m as function of the previously seen URLLCdemands D(1:m−1) := (D(1), D(2), . . . , D(m− 1)).Let d(1:m−1) is a realization of D(1:m−1). For anym and d(1:m−1), γπ,su,m

(d(1:m−1)

)has to satisfy the

following constraints.∑u∈U

γs,πu,m(d(1:m−1)) = 1, (16)

γs,πu,m(d(1:m−1)) ≤ φπ,su|M| (1− δ)

∀u ∈ U , (17)

γs,πu,m(d(1:m−1)) ∈ [0, 1] ∀u ∈ U . (18)

Observe that the URLLC placement factor for causal andminislot-dependent scheduling policy is not just dependent onthe user and network state but it also depends on the mini-slotindex and past URLLC demands.

Let Π be the set of all causal and mini-slot dependentscheduling policies. In our online algorithm (9), for any

8

eMBB slot t, we find the policy which solves the followingoptimization problem with non-negative weights wu.

OP1 : maxπ∈Π

:∑u∈U

wugπ,su (φπ,su ,γπ,su ) , (19)

where s is the current network state, γπ,su :=(γπ,su,1 (·) , γπ,su,2 (·) , . . . , γπ,su,|M| (·)

)is the vector of URLLC

placement factors of all minislots (with slight abuse ofnotation) and gπ,su (·, ·) is the average rate experienced byeMBB user u under policy π. gπ,su (·, ·) is given by thefollowing expression:

gπ,su (φπ,su ,γπ,su ) :=

rsuφπ,su E

[1− hsu

(∑|M|m=1 γ

π,su,m

(D(1:m−1)

)Dm

φπ,su

)], (20)

where the expectation is computed with respect to the jointdistribution of D(1), D(2), . . ., D(|M|). One can formu-late the above optimization problem as a Markov DecisionProblem (MDP), however the state space for such an MDPis prohibitively large. Furthermore we note that minislot-homogeneous policies are attractive in terms of it compu-tational complexity. In general, one cannot expect optimalminislot-homogeneous policies to perform as well as optimalminislot dependent policies, however, if we restrict ourselvesto convex homogeneous loss functions, then we can show thatminislot-homogeneous policies are in fact optimal over Π.

Definition 4. A loss function hsu(·) is said to be homogeneousif there exists a real number p such that ∀ x ∈ [0, 1] and κ ≥ 0we have that

hsu(κx) = κphsu(x). (21)

Even with this restriction we can model useful loss functionswhich could possibly be user and network state dependent.Some examples are given below.

1) Linear: hsu(x) = ksu (x), where ksu ≥ 0.2) Monomial: hsu(x) = ksu (x)

q where ksu ≥ 0 and q ≥ 1.Our main result on the optimality of minislot-homogeneous

policies is proved in Appendix F and stated next.

Theorem 5. If the support of URLLC demands D is a finitediscrete set and eMBB loss functions are homogeneous andconvex, then there exists an optimal solution (φs,∗,γs,∗ (·)) forOP1 with a minislot-homogeneous URLLC placement policyγs,∗.

E. Optimal eMBB Slot Slicing

In Section II we have used uniform slot sizes for eMBBusers, i.e. the allocated minislots to all users span the en-tire width of the slot (see Figure 4; henceforth referred toas frequency slices). However, new proposals allow greaterflexibility in slot allocation, e.g., the capability to choosedifferent slices over both time and frequency for differenteMBB users [1]. In this section we will show that whileit is possible to slice eMBB users’ resources flexibly, it is

Fig. 3. Time Slices: In this configuration, eMBB users share resources overtime in an eMBB slot.

Fig. 4. Frequency Slices: In this configuration, eMBB users homogeneouslyshare frequency in an eMBB slot.

preferable to slice frequency (see 4) than time from the pointof view of puncturing losses for convex loss functions.

The essence of the discussion can be captured by comparingthe two resource allocation configurations shown in Figures 3and 4. In Configuration 1 (time slices), eMBB user 1 isallocated the entire frequency band for a subset of m1 minis-lots. Similarly eMBB user 2 is allocated the entire frequencyband for its subset of m2 minislots. The network state s isassumed to be the same for the entire m1 +m2 minislots. Thisimplies that the loss functions of eMBB users (hsu (·)) do notchange throughout the m1 +m2 minislots. In Configuration 2(frequency slices) we allocate an eMBB user 1 a fraction φ1

of the bandwidth for a duration of m1 +m2 minislots, whereφ1 := m1

m1+m2and similarly for eMBB user 2. Note that the

total resources allocated to eMBB users, which is representedby the area allocated in the time-frequency plane is same inboth configurations.

In Configuration 1, the total puncturing observed by eMBBuser 1 is given by

∑m1

m=1D(m) and similarly for eMBBuser 2. Whereas in Configuration 2, under uniform URLLCplacement, the total puncturing observed by eMBB user 1is given by

∑m1+m2

m=1 φ1D(m). Note that the mean totalpuncturing is same in both the configurations.

The main result of this section is given below:

9

Theorem 6. Under the assumption of i.i.d. URLLC demands5

(D(m), m = 1, 2, . . . ,m1 + m2) and convex loss functions(hsu (·)), for any eMBB user, e.g., eMBB user 1, we have that

E

[hs1

(m1∑m=1

D(m)

)]≥ E

[hs1

(m1+m2∑m=1

φ1D(m)

)]. (22)

Proof of this result is given in Appendix G.Remarks: The above theorem shows that the expected loss

suffered by an eMBB user due to URLLC puncturing inConfiguration 1 (time slicing) is higher than in Configuration 2(frequency slicing). This implies that it is preferable for eMBBusers to spread their resource allocation over time from theperspective of reducing their loss due to puncturing. Theunderlying reason is that Configuration 2 results in smallervariability in the total puncturing even though both the con-figurations have the same mean total puncturing. Since theloss functions are convex, a lower variability leads to a lowerexpected loss. Finally, for more complex (rectangular) slices,we can now apply Thm. 6 iteratively and show that usingfrequency slices with appropriate scaling of the bandwidthallocation results in a higher average rate for eMBB users.

V. THRESHOLD MODEL AND PLACEMENT POLICIES

In the previous section, we developed a stochastic approx-imation based algorithm for minislot-homogeneous policies.This algorithm iteratively solves the optimization problemgiven in (9). This optimization problem jointly optimizes overa pair of row vectors (φs,γs). While this convex optimizationproblem can be solved using standard methods, it couldbecome computationally challenging as the number of usersincreases.

In this section, we shall restrict our attention to a thresholdmodel for superposition/puncturing, and look at policies thatimpose structural conditions on the puncturing matrix γ.We will show that the resulting class of policies have nicetheoretical properties that lead to simpler online algorithms(solving (4), which is an one-dimensional search).

We consider two types of structural conditions on γ:(i) Resource Proportional (RP) Placement: The first is basedon allocating URLLC demands in proportion to eMBB userslot allocations, i.e., γsu = φsu. We refer to this as ResourceProportional (RP) Placement and denote such policies by

ΠRP,δ := {(φ,γ) ∈ ΠH,δ | γ = φ},

and define the associated achievable throughput region

CRP,δ = {c ∈ R|U|+ | ∃π ∈ ΠRP,δ s.t. c ≤ cπ}.

The motivation for RP Placement comes from the optimalityof random placement for the linear model in Section III.Observe that if puncturing occurs uniformly randomly, thenthe expected number of punctures is directly proportional tothe fraction of bandwidth allocated to an eMBB user. Thus,RP Placement can be viewed as a determinized version of the

5This result can be extended to exchangeable URLLC demands. We usei.i.d. assumption to maintain consistency with other sections.

random placement strategy which ensures that the proportionsof puncturing satisfy resource proportional ratios.(ii) Threshold Proportional (TP) Placement: The secondpolicy allocates URLLC demands in proportion to the eMBBusers associated loss thresholds so as to avoid losses,

γsu =φsut

su(φsu)∑

u′∈U φsu′tsu′(φsu′)

.

We refer to this as Threshold Proportional (TP) Placement anddenote such policies by

ΠTP,δ :=

{(φ,γ) ∈ ΠH,δ | γsu =φsut

su(φsu)∑

u′∈U φsu′tsu′(φsu′)

∀s ∈ S, u ∈ U}.

The associated achievable throughput region is denoted

CTP,δ = {c ∈ R|U|+ | ∃π ∈ ΠTP,δ s.t. c ≤ cπ}.

First we state a corollary to Theorem 2 which characterizesthe rates under different URLLC placement policies for sys-tems having threshold loss model for superposition/puncturing.

Corollary 1. Under a (1 − δ) sharing factor and time-homogeneous scheduler π = (φπ,γπ) ∈ ΠH,δ the probabilityof induced eMBB loss for user u ∈ U in channel state s ∈ Sis given by

επ,su = 1− FD(φπ,su tsu(φπ,su )

γπ,su).

where FD denotes the cumulative distribution function of theURLLC demands on a typical eMBB slot. Then the associateduser throughput is given by

rπ,su = rsuφπ,su FD(

φπ,su tsu(φπ,su )

γπ,su).

and the overall user throughputs are given by cπ = (cπu : u ∈U) where

cπu =∑u∈U

rsuφsuFD(

φπ,su tsu(φπ,su )

γπ,su)pS(s).

The following two corollaries are direct consequences ofCorollary 1 and Theorem 3 restricted to RP and TP Placementstrategies, and characterize the capacity regions under the twopolicies.

Corollary 2. Consider a wireless system with full sharingfactor and time-homogeneous scheduler based on the RPURLLC Placement policy π = (φπ,γπ) ∈ ΠRP,δ. Thenany eMBB resource allocation φ combined with a RP URLLCdemand placement policy, γ = φ is feasible. The probabilityof loss for user u ∈ U in channel state s ∈ S is given by

επ,su = 1− FD(tsu(φπ,su )),

with associated user throughput

rπ,su = rsuφsuFD(tsu(φπ,su )). (23)

10

Further if for all s ∈ S and u ∈ U the functions gsu(, ) givenby

gsu(φsu) = φsuFD(tsu(φπ,su )), (24)

are concave then CRP,δ = CRP,δ.

Corollary 3. Under a (1 − δ) sharing factor and jointlyuniform scheduler based on the TP URLLC Placement policyπ = (φπ,γπ) ∈ ΠTP,δ, the probability of induced eMBB lossuser u ∈ U in channel state s ∈ S is given by

επ,su = 1− FD(∑u∈U

φπ,su tsu(φπ,su )), (25)

with associated user throughput

rπ,su = rsuφsuFD(

∑u∈U

φπ,su tsu(φπ,su )). (26)

Further if for all s ∈ S and u ∈ U the functions gsu(, ) givenby

gsu(φsu, γsu) = φsuFD(

∑u∈U

φπ,su tsu(φπ,su )), (27)

are jointly concave then CTP,δ = CTP,δ.

The following theorem provides a formal motivation for TPPlacement. The main takeaway here is that the probabilityof any loss in an eMBB slot under TP Placement policy isa lower bound for all other strategies. Note that minimizingthe probability of any eMBB loss is not same as minimizingeMBB rate loss.

Theorem 7. Consider a system with (1 − δ) sharing factor.Consider a joint scheduling policy based on the TP URLLCplacement i.e, π = (φπ,γπ) ∈ ΠTP,δ. Then π achievesthe minimum probability of any eMBB loss amongst all jointscheduling policies using the same eMBB resource allocationφπ.

The proof is included in Appendix H.Next we consider online algorithms that implement the RP

and TP Placement policies. While the stochastic approximationalgorithm developed in Section IV-C can clearly be used, theadditional structure imposed by the RP and TP Placement poli-cies, and the shape of the threshold loss function (discussedbelow) can result in much simpler algorithms (with optimalityguarantees).

We consider the case where tsu(φ) is a (state dependent butφ independent) constant, i.e., tsu(φ) = αs, where αs ∈ (0, 1).Intuitively, this means that eMBB traffic which has a highershare of the bandwidth is more resilient to losses (e.g. throughcoding over larger fraction of resources). Then, by substitutingthis loss function in (23) and (26) (where we also use the factthat

∑u∈U φ

su = 1), we have that

rπ,su = rsuφsuFD(αs).

Comparing with the development in Section III-B, we observethat the cost and constraints are identical if FD(αs) replaces(1 − ρ). Note that a small difference is that FD(αs) is

state dependent, whereas (1 − ρ) does not depend on thestate; however, it is easy to see that the development inSection III-B immediately generalizes to this setting. Hence,we can interpret FD(αs) as the state dependent average rateloss due to puncturing via the RP or TP Placement policies.

We can now employ the rate-based iterative gradient sched-uler developed in Section III-B (by replacing (1 − ρ) in (5)by a user-dependent FD(αs)), and the theoretical guaranteesdirectly carry over. As this algorithm only minimizes overusers at each slot in (4), this is easier to implement whencompared to the stochastic approximation algorithm developedin Section IV-C.

VI. SIMULATIONS

We consider a system with a total of 100 RBs availableper eMBB slot, and with 8 minislots per eMBB slot. In aneMBB slot, rsu for an eMBB user is drawn from the finiteset {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Mbps according to a probabilitydistribution and i.i.d. across users and slots. Our systemconsists of 20 users, and with 100 channel states (all equallylikely). The (20 users × 100 states) rate matrix is one-timesynthesized by independently and uniformly sampling a ratefrom the finite rate set for each matrix element. For 10 eMBBusers, we have chosen the probability distribution such that theaverage rate is 7 Mbps. For the rest, probability distribution issuch that the average rate is 3 Mbps. This models two classesof users, one class with higher link rates which can toleratea higher amount of puncturing and the other with lower linkrates which can tolerate lesser amount of puncturing. This isreasonable as a user with a higher channel rate can code morerobustly and protect its transmissions from URLLC puncturingmore than a user with a lower channel rate. In this spirit weshall call users with 7 Mbps average rates as ‘robust’ usersand users with 3 Mbps average rates as ‘sensitive’ users. Weuse the utility function Uu(r) = log (r) for all users.

We first show that joint scheduling is necessary to preserveeMBB throughputs. To that end we benchmark our optimalonline algorithm (stochastic approximation algorithm, see Sec-tion IV-C) for convex loss functions with a scheme whichperforms standard gradient based scheduling for eMBB usersand Resource Proportional (RP) URLLC placement. Note thatfor convex loss functions, RP placement strategy does nottake into account the eMBB user’s sensitivity to delays. Forusers with average rate 7 Mbps, we use the loss functionhsu(x) = x2. For users with average rate 3 Mpbs, we usethe following loss function:

hsu(x) =

{(x

0.7

)2, if x ≤ 0.7,

0, if 0.7 < x ≤ 1.(28)

URLLC demands in a minislot is drawn from a binomialdistribution which can take values 0 with p and 1−δ

8 withprobability 1 − p. Note that this ensures that peak URLLCload in an eMBB slot is less than or equal to 1− δ.

In Fig. 5, we compare the average sum utility under ouroptimal joint scheduler and the RP based policy as a functionof the URLLC load. As the load increases, RP performs

11

0.1 0.2 0.3 0.4 0.5 0.6 0.7Average URLLC load ( )

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

Sum

util

ity

Optimal algorithmRP

Fig. 5. Sum utility as a function of URLLC load ρ for the optimal and RPpolicies under convex model δ = 0.3.

poorly. To understand this phenomenon in detail, we haveplotted the average rates of robust and sensitive users underthe two policies in Fig. 6. As we increase the URLLC load, theaverage eMBB rates of both sensitive and robust users decreaserapidly. For example, when ρ = 0.4, RP has 15 % lowerthroughput for robust users and almost similar performancefor sensitive users as compared to optimal algorithm. Furtheras we increase ρ to 0.6, the throughput of robust and sensitiveusers in RP decrease by 35 % and 26 %, respectively.

Sensitive users are the most affected by URLLC puncturing.When the RP URLLC placement policy is combined with thestandard gradient based algorithm for eMBB users, it allocatesmore resources to sensitive users because they have highermarginal utility. Since sensitive users receive more bandwidth,under the RP URLLC placement strategy they receive morepuncturing. This will lead to even more allocation of resourcesto sensitive users and this process continues until robustusers have similar marginal utilities (due to reduced rates) assensitive users. Hence, the robust users are resource starved.As we increase the URLLC load further, sensitive users receiveeven more URLLC puncturing and neither the robust norsensitive users get good average rates when compared tothe optimal joint scheduler. This shows that we require jointscheduling of eMBB and URLLC to exploit the heterogeneityin sensitivities to URLLC puncturing in maximizing eMBButilities.

Next we consider a threshold based loss model with αs =0.3 for 50% of eMBB states and αs = 0.7 for the rest.We use the utility function Uu(r) = log(r) + 6.5 for alleMBB users, where r is measured in Mbps (constant addedto ensure non-negativity of the sum utility). URLLC loadin an eMBB slot (D) is generated based on the truncatedPareto distribution with tail exponent η = 2. We comparethe optimal policy (stochastic approximation algorithm, seeSection IV-C) with that from the TP Placement policy (thesimpler gradient algorithm in Section V). In this case, sincethe threshold functions are (state-dependent) constants, the RPand TP Placement policies are the same. As we can see inFigure 7, unlike the convex loss model the RP/TP Placement

0.1 0.2 0.3 0.4 0.5 0.6 0.7Average URLLC load ( )

0

0.1

0.2

0.3

0.4

0.5

0.6

Aver

age

rate

Robust users-- Opt. algorithmSensitive users-- Opt. algorithmRobust users-- RPSensitive users-- RP

15 %

35 %

26 %

Fig. 6. Average rates as a function of URLLC load ρ for the optimal and RPpolicies under convex model δ = 0.3.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Average URLLC Load

02

12

22

32

42

Sum

Utilit

y

RP/TP

Optimal joint scheduler

Fig. 7. Sum utility as a function of URLLC load ρ for the optimal and TPPlacement policies under threshold model (δ = 0.1).

policy tracks the optimal policy very well.In Figure 9, we study the trade-off between achieving a

higher eMBB utility and lowering the mean delay of URLLCtraffic for different values of the sharing factor 1−δ. Figure 9plots the corresponding probability that the URLLC trafficdelay exceeds two minislots (0.125×2 = 0.25 msec). To studythis trade-off we generate URLLC arrivals in each minislotfrom an uniform distribution between [0, 1/8] (recall there are

0.1 0.15 0.2 0.25 0.3 0.35 0.41

2

3

4

5

6

7

8

Sum

util

ity

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Mea

n de

lay

(min

islo

ts)

Sum utilityMean delay

Fig. 8. Sum utility and mean URLLC delay as a function of δ.

12

0.25 0.3 0.35 0.410-8

10-6

10-4

10-2

100

De

lay p

rob

ab

ility

Fig. 9. Log-scale plot of the probability that URLLC traffic is delayed bymore than two minislots (0.25 msec) for various values of δ.

8 minislots). In each minislot, we can serve at most 1−δ8 units

of URLLC traffic. If the URLLC load in a given minislot ismore than 1−δ

8 , the remaining URLLC traffic is queued andserved in the next minislot on a FCFS basis. For the eMBBusers we use a convex model with hsu(x) = eκu(x−1) whereκu determines the sensitivity of an eMBB user to an URLLCload. We have chosen κ = 0.2 for 50 % of the users andκ = 0.7 for the rest. We also set ∀u Uu(x) = log(x) + 4.2(constant added to ensure positive sum utility). In summary,a larger value of δ limits the amount of URLLC traffic thancan be served in a minislot. However, a larger δ enlarges theconstraint set ΠH,δ in the eMBB utility maximization problem,and hence we get higher eMBB utility.

VII. CONCLUSION

In this paper, we have developed a framework and al-gorithms for joint scheduling of URLLC (low latency) andeMBB (broadband) traffic in emerging 5G systems. Oursetting considers recent proposals where URLLC traffic isdynamically multiplexed through puncturing/superposition ofeMBB traffic. Our results show that this joint problem hasstructural properties that enable clean decompositions, andcorresponding algorithms with theoretical guarantees.

ACKNOWLEDGEMENTS

The work of Arjun Anand was partially supported byFutureWei Technologies and NSF grant CNS-1731658, Gus-tavo de Veciana was partially supported by NSF grantsCNS-1343383 and CNS-1731658, and Sanjay Shakkottai waspartially supported by NSF grants CNS-1343383 and CNS-1731658, and the US DoT D-STOP Tier 1 University Trans-portation Center.

REFERENCES

[1] 3GPP TSG RAN WG1 Meeting 87, November 2016.[2] A. Anand, G. de Veciana, and S. Shakkottai, “Joint scheduling of

URLLC and eMBB traffic in 5g wireless networks,” in Proc. INFOCOM,May 2018.

[3] K. I. Pedersen, G. Berardinelli, F. Frederiksen, P. Mogensen, andA. Szufarska, “A flexible 5G frame structure design for frequency-division duplex cases,” IEEE Communications Magazine, vol. 54, no. 3,pp. 53–59, March 2016.

[4] Chairman’s notes 3GPP: 3GPP TSG RAN WG1 Meeting 88bis,Available at http://www.3gpp.org/ftp/TSG RAN/WG1 RL1/TSGR188b/Report/, April 2017.

[5] R. Srikant and L. Ying, Communication Networks: An Optimization,Control, and Stochastic Networks Perspective. Cambridge UniversityPress, 2014.

[6] L. Georgiadis, M. J. Neely, and L. Tassiulas, “Resource allocation andcross-layer control in wireless networks,” Foundations and Trends inNetworking, vol. 1, no. 1, 2006.

[7] B. Holfeld, D. Wieruch, T. Wirth, L. Thiele, S. A. Ashraf, J. Huschke,I. Aktas, and J. Ansari, “Wireless communication for factory automa-tion: an opportunity for LTE and 5G systems,” IEEE CommunicationsMagazine, vol. 54, no. 6, pp. 36–43, June 2016.

[8] O. N. C. Yilmaz, Y. P. E. Wang, N. A. Johansson, N. Brahmi, S. A.Ashraf, and J. Sachs, “Analysis of ultra-reliable and low-latency 5Gcommunication for a factory automation use case,” in 2015 IEEEInternational Conference on Communication Workshop (ICCW), June2015, pp. 1190–1195.

[9] M. Gidlund, T. Lennvall, and J. Akerberg, “Will 5G become yetanother wireless technology for industrial automation?” in 2017 IEEEInternational Conference on Industrial Technology (ICIT), March 2017,pp. 1319–1324.

[10] C.-P. Li, J. Jiang, W. Chen, T. Ji, and J. Smee, “5G ultra-reliable andlow-latency systems design,” in 2017 European Conference on Networksand Communications (EuCNC), June 2017, pp. 1–5.

[11] G. Durisi, T. Koch, and P. Popovski, “Toward massive, ultrareliable, andlow-latency wireless communication with short packets,” Proceedings ofthe IEEE, vol. 104, no. 9, pp. 1711–1726, Sept 2016.

[12] G. Durisi, T. Koch, J. Ostman, Y. Polyanskiy, and W. Yang, “Short-packet communications over multiple-antenna rayleigh-fading channels,”IEEE Trans. on Comm., vol. 64, no. 2, pp. 618–629, Feb 2016.

[13] B. Singh, Z. Li, O. Tirkkonen, M. A. Uusitalo, and P. Mogensen,“Ultra-reliable communication in a factory environment for 5G wirelessnetworks: Link level and deployment study,” in 2016 IEEE 27th An-nual International Symposium on Personal, Indoor, and Mobile RadioCommunications (PIMRC), Sept 2016, pp. 1–5.

[14] L. You, Q. Liao, N. Pappas, and D. Yuan, “Resource Optimization withFlexible Numerology and Frame Structure for Heterogeneous Services,”ArXiv e-prints, 2018.

[15] P. Popovski, K. F. Trillingsgaard, O. Simeone, and G. Durisi,“5G Wireless Network Slicing for eMBB, URLLC, and mMTC: ACommunication-Theoretic View,” ArXiv e-prints, 2018.

[16] R. Kassab, O. Simeone, and P. Popovski, “Coexistence of URLLC andeMBB services in the C-RAN Uplink: An Information-Theoretic Study,”ArXiv e-prints, 2018.

[17] D. Julian, “Erasure networks,” in Proceedings IEEE International Sym-posium on Information Theory,, Jul. 2002.

[18] A. L. Stolyar, “On the asymptotic optimality of the gradient schedulingalgorithm for multiuser throughput allocation,” Operations Research,vol. 53, no. 1, pp. 12–25, 2005.

[19] S. Bodas, S. Shakkottai, L. Ying, and R. Srikant, “Low-complexityscheduling algorithms for multi-channel downlink wireless networks,”in Proceedings of IEEE Infocom, 2010.

[20] ——, “Scheduling for small delay in multi-rate multi-channel wirelessnetworks,” in Proceedings of IEEE Infocom, 2011.

[21] S. Boyd and L. Vandenberge, Convex Optimization. CambridgeUniversity Press, 2003.

[22] H. J. Kushner and G. G. Yin, Stochastic Approximation Algorithms andApplications. Springer, 1997.

13

APPENDIX

A. Proof of Theorem 1

Clearly since ΠLR ⊂ Π we have that CLR ⊂ CNow consider any policy π ∈ Π with eMBB user allocations

φπ and URLLC loads lπ

and associated long term throughputis cπ given by

cπu =∑s∈S

rsu(φπ,su − lπ,s

u )pS(s).

Let us define a π′ based on π to have per minislot eMBBuser allocations given by

φπ′,su,m =

φπ,su − lπ,s

u∑u′∈U φ

π,su′ − lπ,su′

f =φsu − l

π,s

u

1− ρf,

for s ∈ S, u ∈ U and m ∈ M. Since induced mean loadson an eMBB user can not exceed its allocation we have thatφπ ≥ l

πso the above allocations are positive. Note also that

this allocation is not minislot dependent, but normalized sothat per minislot they sum to f and over the whole eMBBslot sum to 1, i.e., φπ

′ ∈ Σ. Thus for such an allocation wehave that

φπ′,su =

φsu − lπ,s

u

1− ρ.

Also suppose that π′ uses randomized URLLC placementacross minislots which induces mean URLLC loads propor-tional to the allocations, i.e., l

π′,s

u = ρφπ′,su . It follows that

φπ′,su − lπ

′,s

u = φπ′,su − ρφπ

′,su

= (1− ρ)φπ′,su

= φπ,su − lπ,s

u ,

and so cπ,su = cπ′,su for all s ∈ S and u ∈ U . Thus for any

policy π there is a policy π′ which uses randomized URLLCplacement and achieves the same long term throughputs. Itfollows that C ⊂ CLR and so C = CLR.

B. Proof of Theorem 2

Under a policy π = (φπ,γπ) ∈ ΠH,δ we have that theinduced loads are given by

Lπ,su,m =γπ,su,1

fD(m),

so we have that

Lπ,su =∑u∈U

Lπ,su,m =γπ,su,1

f

∑u∈U

D(m) =γπ,su,1

fD = γπ,su D.

where the last equality follows from the uniformity of URLLCsplits and normalization it follows that

rπ,su = E[fsu(φπ,su , Lπ,su )] = E[fsu(φπ,su , γπ,su D)].

C. Proof of Lemma 1

Recall that convex loss functions are specified as follows

fsu(φsu, lsu) = rsuφ

su(1− hsu

(lsuφsu

)),

with hsu : [0, 1] → [0, 1] a convex increasing function. Fortime-homogenous policies we have defined

gsu(φsu, γsu) = E[fsu(φsu, γ

suD)]

= rsuE[φsu − φsuhsu(γsuφsuD)].

Recall that convex function h(·) one can define a functionl(φ, γ) = φh( γφ ) known as the perspective of h(·) which isknown to be jointly convex in its arguments. It follows thatφ − φh( γφ ) is jointly concave, and so is gsu(·) since it is aweighted aggregation of jointly concave functions.

For threshold-based loss functions where tsu(φsu) = αsu wehave that

gsu(φsu, γsu) = E[fsu(φsu, γ

suD)]

= rsuφπ,su P (γsuD ≤ φπ,su αus )

= rsuφπ,su FD(

φπ,su αusγsu

).

Now using the same result on the perspective functions ofvariables the result follows. The truncated Pareto case can beeasily verified by taking derivatives.

D. Proof of Theorem 3

Clearly CH,δ ⊂ CH,δ. We will show that c ∈ CH,δ then theirexists π = (φπ,γπ) ∈ ΠH,δ such that c ≤ cπ from which itfollows that CH,δ ⊂ CH,δ.

Suppose c ∈ CH,δ , then it can be represented as a convexcombination of policies ΠH,δ , in each channel state. Forexample suppose for simplicity that for that in channel states ∈ S we have that λ ∈ [0, 1] one time shares between twopolicies π1 and π2 to achieve throughputs for u ∈ U givenby

rsu = λrπ1,su + (1− λ)rπ2,s

u .

Consider u we have

rsu = λrπ1,su + (1− λ)rπ2,s

u

= λgsu(φπ1,su , γπ1,s

u ) + (1− λ)gsu(φπ2,su , γπ2,s

u )

≤ gsu(λφπ1,su + (1− λ)φπ2,s

u , λγπ1,su + (1− λ)φγ2,su )

= gsu(φπ,su , γπ,su ),

where φπ,su = λφπ1,su + (1 − λ)φπ2,s

u and γπ,su = λγπ1,su +

(1− λ)γπ2,su . Clearly φπ,γπ as given above correspond to a

policy π such that π ∈ ΠH,δ since the set is convex. It alsofollows that rsu ≤ rπ,su , so csu ≤ cπ,su and so c ≤ cπ.

14

E. Proof of Theorem 4

The proof requires intermediate lemmas, de-tailed below. For the ease of exposition, let usdefine U(r) :=

∑u∈U Uu(ru) and ∇U (r) :=(

∂U1(x)∂x

∣∣∣x1=r1

, ∂U2(x)∂x

∣∣∣x2=r2

, . . . , ∂U1(x)∂x

∣∣∣x|U|=r|U|

)T.

First we have the following important lemma regarding thestochastic approximation algorithm.

Lemma 2. R(t) =(R1(t), R2(t), . . . , R|U|

)Tis an unbiased

estimator of argmax:c∈CH,δ

∇U(R(t)

)Tc, i.e.,

E [R(t)] = argmax:c∈CH,δ

∇U(R(t)

)Tc. (29)

Proof. Based on the definition of CH,δ we can re-writemax:c∈CH,δ

∇U(R(t)

)Tc as follows:

maxφ,γ

∑u∈U

U′

u

(Ru(t)

)(∑s∈S

pS(s)gsu (φsu, γsu)

), (30)

s.t. φ ≥ (1− δ)γ, (31)

φ, γ ∈ ΠH,δ. (32)

Observe that the above optimization problem can be solvedseparately for each network state s ∈ S . The de-coupledproblem for any state s is same as the optimization problem (9)in our online algorithm. With a slight abuse of notation, let(φ(s), γ(s)

)be the optimal solution to the online problem

when S(t) = s. Conditioned on S(t) = s, we have that:

E [Ru(t) | S(t) = s] = E[fsu

(φsu, γ

suD)| S(t) = s

]= gsu

(φsu, γ

su

)∀u ∈ U . (33)

Computing E [E [Ru(t) | S(t)]] gives the desired result (29).

The main intuition behind the proof of optimality is thatfor large t, the trajectories of R(t) can be approximated bythe solution to the following differential equation in x(t) withcontinuous time t:

dx(t)

dt= argmax:

c∈CH,δ∇U (x(t))

Tc− x(t). (34)

Let us define q(x) := argmax:c∈CH,δ

∇U (x)Tc. To show the

optimality of our online algorithm, we shall also require thefollowing result on the above differential equation.

Lemma 3. The differential equation (34) is globally asymp-totically stable. Furthermore, for any initial condition x(0) ∈CH,δ , we have that limt→∞ x(t) = r∗.

Proof. To prove this lemma it is enough to show that thereexists a Lyapunov function L(x(t)) such that it has a negativedrift when x(t) 6= r∗ and has zero drift when x(t) = r∗.Define L(x) = U(r∗) − U(x). Observe that under ourassumption of strictly concave Uu(·), the offline optimization

problem is guaranteed to have an unique optimal solution,which is r∗. Therefore, ∀x ∈ CH,δ and x 6= r∗ L(x) > 0.Next we will compute the drift of L(x(t)) with respect totime.

dL(x(t))

dt= −∇U (x(t))

T dx(t)

dt, (35)

= −q (x(t)) +∇U (x(t))Tx(t), (36)

< 0 ∀x(t) 6= r∗. (37)

To get inequality (37), first observe that from the definitionof q(x(t)) and (36), we get that dL(x(t))

dt ≤ 0. However,we have to show that this inequality is strict for x(t) 6= r∗.Observe that q(x) = x is a necessary and sufficient conditionfor optimality of the offline optimization problem, see [21] formore details. From strict concavity of the utility functions, wehave an unique optimal point r∗. Therefore, dL(x(t))

dt < 0 forx(t) 6= r∗ and dL(x(t))

dt = 0 at x(t) = r∗.

To conclude the proof, Lemmas 2 and 3 along with thecondition 3 satisfy all the conditions necessary to apply The-orem 2.1 in Chapter 5, [22] which states that R(t) convergesto r∗ almost surely.

F. Proof of Theorem 5

The proof has the following two steps.1) We shall first consider a hypothetical non-casual sce-

nario and show that there exists an optimal joint schedul-ing policy with minislot-homogeneous URLLC place-ment policy which in general is a function of theaggregate URLLC load in an eMBB slot. We then upperbound the optimal value of OP1 by the solution to ahypothetical non-causal scenario described in the sequel.

2) Secondly, under Assumption 4 on the loss functions,we show that there exists an URLLC placement policypolicy which is minislot-homogeneous but independentof the aggregate URLLC load for the hypothetical non-causal scenario. We then conclude that there exists anoptimal minislot-homogeneous joint sceduling policy forOP1 as an upper bound for its value is attained by aminislot-homogeneous joint scheduling policy.

The two steps are elaborated next.1) Hypothetical non-causal scenario: First let us describe

the non-causal scenario. At the beginning of each eMBBslot, first the scheduler chooses φπ,s. Next the total URLLCdemand in each minislot is revealed, i.e., the realizations ofD(1), D(2), . . . , D(|M|) are revealed. Therefore, this settingis not causal as it assumes knowledge about future URLLCdemand realizations. In general the URLLC placement underthe non-causal setting is dependent on the minislot index mand D(1:|M|). With slight abuse of notation, we shall denote itby γsu,m

(D(1:|M|)). The joint scheduling policy has to satisfy

the constraints (15), (16), and (17). We have the followinglemma on the non-causal setting.

Lemma 4. There exists an optimal minislot-homogeneous pol-icy for the non-casual setting such that the URLLC placement

15

depends only on the total URLLC demand in an eMBB slot,i.e.,

∑|M|m=1Dm.

Proof. Let(φπ, γπ,s (·)

)be the decision variables un-

der an optimal joint scheduling policy π in the non-causal setting. Let d(1), d(2), . . . , d(|M|) be realizations ofD(1), D(2), . . . , D(|M|) such that

∑|M|m=1 d(m) = d. Define

the following:

νsu :=

∑|M|m=1 γ

π,su,m

(d(1:|M|)) d(m)

d. (38)

Note that with the definition of νsu, the total puncturingexperienced by an eMBB user u in an eMBB slot isνsud. From this one can construct an equivalent minislot-homogeneous URLLC placement policy. For all minislots,use νs as the URLLC placement factor. This satisfies theconstraints (15), (16), and (17). In general νs could dependon d(1), d(2), . . . , d(|M|). However, we will show that theoptimal solution depends only on the sum

∑|M|m=1 dm.

Let d′(1), d′(2), . . . , d′(|M|) be such that∑|M|m=1 d

′m = d

and there exists an m such that d′(m) 6= d(m). Define thefollowing:

ν′su :=

∑|M|m=1 γ

π,su,m

(d′(1:|M|)) d′md

. (39)

Therefore, the total puncturing observed by ν′su d. Observethat ν′s is also a feasible URLLC policy for the case whenthe URLLC demand realizations are d(1), d(2), . . . , d(|M|).Similarly νs is also a feasible URLLC placement policyfor the case with d′(1), d′(2), . . . , d′(|M|). Therefore, theoptimal solution has to be independent of the realizationsof D(1), D(2), . . . , D(|M|) and depends only on the sum∑|M|m=1Dm.

Therefore, we shall restrict ourselves to minislot-homogeneous policies in the non-causal setting with theURLLC placement as a function of the total URLLC demandfor that eMBB slot. With slight abuse of notation we shalldenote a URLLC placement policy in this setting by γsu (·)with the only argument as the total URLLC demand in thateMBB slot. This procedure is formally described next.

1) At the beginning of an eMBB slot, the joint schedulerchooses φπ,su , u ∈ U such that∑

u∈Uφπ,su = 1 and φπ,su ∈ [0, 1] ∀u. (40)

2) The total URLLC demand D =∑|M|m=1D(m) in that

eMBB slot is revealed.3) For an URLLC demand of D, γπ,su (D) is chosen such

that ∑u∈U

γπ,su (D) = 1, and γπ,su (D) ∈ [0, 1] . (41)

Let us denote the feasible policies for this hypothetical non-causal scenario by Π†. (φπ,s,γπ,s) is chosen as the solutionto the following optimization problem.

OP2 : maxπ∈Π†

:∑u∈U

wugπ,su (φπ,su , γπ,su (·)) , (42)

where gπ,su (φπ,su , γπ,su (·)) = rsuφπ,su E

[1− hsu

(γπ,su (D)Dφπ,su

)].

We have the following important lemma which states that theoptimal value under the non-causal scenario is an upper boundto the optimal value under the causal and minislot-dependentpolicy.

Lemma 5.

maxπ∈Π†

:∑u∈U

wugπ,su (φπ,su , γπ,su (·))

≥ maxπ∈Π

:∑u∈U

wugπ,su (φπ,su ,γπ,su ) . (43)

Proof. This directly follows from the proof of Lemma 4 wherewe have shown that any URLLC placement factor γπ,su can betransformed into a minislot-homogeneous policy which dependonly on the total URLLC demand in an eMBB slot, and hence,any feasible solution for OP1 is a feasible solution for OP2.

2) Existence of an optimal solution independent of the valueof D: In general the optimal URLLC placement policy underOP2 may depend on the total URLLC demand in an eMBBslot. However, under the Assumption 4 it is independent of thetotal URLLC demand. This is stated formally in the followinglemma.

Lemma 6. Under Assumption 4, there exists an optimal so-lution (φ∗,s,γ∗,s (·)) for OP2 with URLLC placement policy(γ∗,s (·)) independent of D.

Proof. If (φ∗,s,γ∗,s (·)) is an optimal solution to OP2, thenγ∗,s (·) must also be an optimal solution to the followingoptimization problem in γs :=

(γs1(·), γs2(·), . . . , γs|U|(·)

).

maxγs

∑u∈U

wugsu(φ∗,su , γsu (·)), (44)

s.t. φ∗,su ≥ (1− δ) γsu(d) ∀u, d, (45)∑u∈U

γsu(d) = 1 and γsu(d) ∈ [0, 1] ∀u, d. (46)

(47)

For any d and u, from the K.K.T. conditions for the aboveoptimization problem, we have that

−wursudphs′

u

(γ∗,su (d)

φ∗,su

)+β(d)+ηu(d)−νu(d)−λu(d) = 0.

(48)

16

where hs′

u (x) =dhsu(y)dy

∣∣∣y=x

,β(d) is an arbitrary constant(function of d) and ηu(d), νu(d) and λu(d) are constants suchthat

λu(d) (φ∗,su (d)− γ∗,su (1− δ)) = 0 and λu(d) ≥ 0 ∀u,(49)

ηu(d)γ∗,su (d) = 0 and ηu(d) ≥ 0 ∀u,(50)

νu(d) (1− γ∗,su (d)) = 0 and νu(d) ≥ 0 ∀u.(51)

Note that we have used the fact that for a homogeneous lossfunctions hs

′

u (dx) = dphs′

u (x). For any d 6= d, if we chooseβ(d) = β(d) d

p

dp , ηu(d) = ηu(d) dp

dp , νu(d) = νu(d) dp

dp , andλu(d) = λu(d) d

p

dp , then from (48) γ∗,su (d) and φ∗,su satisfy theK.K.T. condition for d

−wursudphs′(γ∗,su (d)

φ∗,su

)+β(d)+ηu(d)−νu(d)−λu(d) = 0.

(52)Hence, γ∗,su (d) and φsu are optimal for d too. Hence, we have aconstructed an optimal solution with URLLC placement policyindependent of D.

We have shown in Lemma 6 that there exists an optimalpolicy (φ∗,s,γ∗,s) which is a minislot-homogeneous policyand independent of the realization of D. In Lemma 5, wehave also shown that the optimal value of OP2 is an upperbound for OP1. Hence, there exists a minislot-homogeneouspolicy which achieves an upper bound for OP1. Therefore,there exists a minislot-homogeneous policy which is optimalfor OP1.

G. Proof of Theorem 6

Let Sk be the set of all subsets with k elements chosen fromthe set {1, 2, . . . ,m1 +m2}. For example, if m1+m2 = 3 andk = 2, then Sk = {{1, 2} , {2, 3} , {1, 3}}. Note that |Sk| =(|M|k

). Using the above definitions, we can re-write the R.H.S.

of (22) as follows:

E

[hs1

(m1+m2∑m=1

φ1D(m)

)]

= E

hs1 1(

m1+m2

m1

) ∑q∈Sm1

(∑m∈q

D(m)

) . (53)

Using the above expression one can apply Jensen’s inequalityon the R.H.S. of (22), we have that

E

[hs1

(m1+m2∑m=1

φ1D(m)

)]

≤ 1(m1+m2

m1

) ∑q∈Sm1

E

[hs1

(∑m∈q

D(m)

)]. (54)

Since Dm’s are i.i.d. the R.H.S. of the above expression issame as the L.H.S. of (22). Hence, proved.

H. Proof of Theorem 7

Clearly the probability of loss depends on the minislotdemands and the users thresholds. If one relaxes the sequentialconstraint on URLLC allocations, one can consider aggregat-ing the the minislot demands and pooling together the userssuperposition/puncturing thresholds. The probability of lossfor this relaxed system is simply the probability the demandexceeds the size of the superposition/puncturing pool, i.e., Theprobability of loss under the pooled resources is given by

P (D ≥∑u∈U

φsutsu(φsu)).

This is clearly a lower bound for any placement policy. Notehowever that the threshold proportional strategy meets thisbound from Corollary 3 (see Equation (25)) so it indeedminimizes the probability of loss on a given eMBB slot.

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