I Introduction
Modelling the locations of BSs and users by Poisson point processes (PPPs), it is shown in [1, 2] that an adaptive transmission scheme based on physical layer rateless codes is very robust in terms of providing enhanced coverage and rate relative to the one based on fixed-rate coding and power control. In [2]
, the metric used to compare the performance of adaptive transmission schemes is the typical user coverage probability and rate, which corresponds to the spatial average of users performance across the network. The meta-distribution of SIR is the distribution of coverage probability in the network conditioned on the point process
[3] [4]. It gives a fine-grained statistical probe into the per-user performance in the network. For the first time in the literature, and using an accurate approximation, this letter characterizes the distribution of the per-user rate, i.e., rate conditioned on the point process in cellular downlink when rateless codes are used. The rate distribution provides a comprehensive probe into the performance enhancements due to rateless coding.Ii Performance Analysis
Ii-a System Model
We consider a single tier cellular downlink in which the locations of BSs are modeled by a PPP , where is a homogeneous PPP of intensity [5]. A user served by a BS is located uniformly at random within the Voronoi cell of . The typical user is located within the typical cell, the Voronoi cell of the BS at origin. The distance between the typical user and the typical BS of is . Its approximate distribution is Rayleigh, with the scale parameter [1]. We consider a translated version of the PPP so that the typical user is at the origin.
On the downlink, each BS transmits a -bit packet to its user using a physical layer rateless code. Each BS transmits with constant power . The channel is quasi-static flat fading affected by path loss. The interference power and SIR at the typical user based on the typical BS transmission are given by
(1) |
(2) |
where and have Exp(1) distribution.
Each packet transmission of bits has a delay constraint of channel uses. The time to decode a -bit packet and thus, the packet transmission time are given by
(3) | |||
(4) |
where is the achievable rate of the typical BS transmission and depends on the type of receiver used. Now, we focus on a framework introduced in [3] to study the network performance conditioned on the PPP . In this letter, the two metrics used to quantify the performance of rateless codes are the success probability and rate of -bit packet transmission conditioned on , defined as
(5) | ||||
(6) |
Both and depend on the distribution of conditioned on . in (6
) is a random variable (RV). It quantifies the per-user rate achieved in a given PPP realization
.In the following, we discuss the conditional packet transmission time distribution for two types of interference models. One type is the constant interference model and the second type is the time-varying interference model.
Ii-B Constant Interference
Under the constant interference (CI) model, it is assumed that the interfering BSs are transmitting continuously to their users for the entire duration of the typical user reception time. The typical BS transmits a -bit packet to the typical user and the performance under such a scenario is characterized below. Note that the CDF in (8) is a RV due to conditioning on . A direct calculation of the distribution of
is infeasible. Below, we quantify the moments of
in (8).Theorem 1.
The moments of the conditional CDF of the packet transmission time, in (8), are given by
(9) |
where is the Gauss hypergeometric function and .
Ii-C Time-varying Interference
Under the time-varying interference (TvI) model, we assume that the interfering BSs transmit only a -bit packet to their user and become silent afterwards. Since the packet transmission time of each interfering BS is random, the interference at the typical user is time-varying. The time-averaged interference up to time is given by [1]
(11) |
where is the packet transmission time of BS .
Assuming each user employs a nearest-neighbour decoder, the achievable rate at the typical user is given by
(12) |
where is based on in (11) and obtained from (2) with replacing . The packet transmission time distribution depends on the Laplace transform (LT) of the interference . However, due to the fact that the marks are correlated, it is infeasible find the LT of . For the sake of analysis, we consider an approximation termed independent thinning model (ITM) in which the correlated marks are replaced by i.i.d. marks with a given CDF . The average interference up to time under ITM is given by
(13) |
Let . We just use for simplicity. The corresponding achievable rate is obtained from (12) with in (13) replacing . The analysis of the TvI model is based on the ITM assumption throughout the paper. The CDF of when conditioned on under the ITM is given by
(14) |
The moments of the RV in (14) are quantified below.
Theorem 2.
The moments of the conditional CDF of the packet transmission time under the independent thinning model, in (14), are bounded as
(15) | ||||
(16) | ||||
(17) | ||||
(18) |
Proof:
Note that in (14) can be written as
(19) | ||||
(20) |
where follows by applying Jensen’s inequality for convex functions in (19). Letting , we get
(21) | |||
(22) |
where (a) follows from the PGFL of the uniform PPP , (b) is obtained from a substitution and the operation w.r.t leads to (c). The function can be expressed as in (22). The relation between in (22) and in (15) is derived in [2, Theorem 1]. ∎
Note that the above function was based on the lower bound in (20). The corresponding function that can be obtained based on the exact expression in (19) is given below.
(23) | ||||
(24) |
in (23) can be further written out in closed form. However, it is not feasible to provide a simple expression for in terms of the hypergeometric function similar to . Hence, we work with the bound for tractability. From now onwards, we just use the tractable approximation defined in (24) instead of . Based on the moments , an expression for the CDF (or its bounds) of the RV in (14) can be written out in closed form[3]. However, since
is supported on the interval [0,1], the beta distribution will be useful. Since
and hence for the CI model, it can be regarded as a special case of the TvI model.Iii Distribution Approximations
Iii-a Beta Approximation of for CI and TvI models
The PDF of beta-approximated is given by
(25) |
where is the beta function and , with and related to the moments of as [3]
(26) |
Note that both the parameters and are functions of .
Iii-B Distribution Approximation for CI and TvI models
From (6), the distribution of can be obtained from the PDF of given in (25) and (26) with and also, the distribution of . Let
(27) |
Now, the CCDF of rate in (6) is given by
(28) |
where is the upper incomplete beta function. To evaluate (28), the distribution of in (27) is very critical. We first obtain the moments of .
From (27), the first two moments of are given by
(29) | ||||
(30) |
where in (a), the second term is given below.
(31) |
where (b) is obtained based on the following steps.
(32) | |||
(33) |
where and . The result in (32) and (33) is derived using steps similar to that in the proof of Theorem 2. The expression is given in (19) and (20). The in (32) is computed using steps similar to (21)-(22), with the exception that is the corresponding function instead of . Using (31) and (33), the second moment in (30) can be computed. For the CI model, the moment expressions are exact. However for the TvI model, the moment expressions are approximate since they are based on the tractable approximation of in (20). It is not feasible to express moments in closed form based on the exact in (19). The PDF of is approximated by a known distribution, whose parameters are expressed in terms of the moments of .
Beta distribution has been used widely to approximate the distribution of a RV with finite support. Hence, we propose to model as a beta distributed RV. The first two moments of are given by
(34) |
Now, the two parameters and of the beta distribution for the RV are given by
(35) |
The PDF of is similar in form to (25) except for the parameters and . Below, we summarize the main result of the preceding discussion.
Theorem 3.
The CCDF of the per-user rate in (6) is given by
(36) |
Iii-C Fixed-Rate Coding
Iii-D Fixed-rate Adaptive Modulation and Coding
Adaptive modulation and coding (AMC) is a scheme used in current 4G networks to adapt the rate to channel conditions[6, 7, 8]. AMC chooses a fixed-rate code and its code rate to most closely match the channel conditions. The preceding discussions in the current paper focused on rateless codes are also applicable to the case of AMC with one important change. For the rateless case, the packet time whereas for the AMC case, the resolution changes to , where is the number of parity symbols for the AMC index .
Iv Numerical Results
In this section, numerical results showing the efficacy of the proposed per-user performance analysis are presented. For the network simulation, the following parameters were chosen: and . In Fig. 1, a plot of the CCDF of the per-user rate is shown for a cellular network at and . The focus is on comparing three types of transmission schemes, i.e., fixed-rate coding based on (38), AMC as described in Section III-D and the rateless scheme as per (36). Note that the performance of AMC is based on simulation only. In terms of matching the rate to the instantaneous channel conditions, fixed-rate coding has poor efficiency and rateless coding has high efficiency with AMC performance lying in between the two schemes. High efficiency of rateless coding is captured by the term in the expression for in (6). For fixed rate coding, the packet time is fixed to . As noted in Section III-D, AMC is also based on fixed-rate codes and thus, the packet time does not change with a finer resolution as compared to the case for rateless coding.
Fig. 2 shows a plot of the CCDF of the per-user rate for a cellular downlink at and . For AMC, the CCDF curve decays to zero at a rate of . For rateless coding, the CCDF at is . Since rateless codes have robust adaptivity to the instantaneous channel conditions, the scheme yields much higher per-user rates relative to the AMC. These higher per-user rates for the rateless scheme have implications on the energy-efficiency of the BS-UE links and also, the congestion, QoS and end-to-end delay in the network. The AMC and rateless scheme have their implementation differences. AMC relies on pre-determined codeword lengths whereas the rateless scheme does not.
Enhancing the accuracy of the analysis: In Section III-B, the key ingredients in obtaining a distribution approximation for the per-user rate were the moments and distribution of . Currently, we have restricted ourselves to the first two moments of . It is expected that using the three moments of leads to a better approximation of its distribution and thus, the same for distribution. However due to space limitations, we have not considered such an approach.
V Conclusion
In this letter, we characterize the per-user performance of cellular downlink when physical layer rateless codes are used for adaptive transmission. The BS locations are modeled by a Poisson point process. Performance of rateless codes was presented under both the constant and time-varying interference models, respectively. Accurate approximations to the distribution of the per-user coverage probability and transmission rate are derived. The advantages of rateless codes are clearly illustrated by comparing their performance to fixed-rate based adaptive modulation and coding.
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