---
title: Rolling-Round-Robin Rate Overview
url: https://www.emergentmind.com/topics/rolling-round-robin-rate
type: topic
---

# Rolling-Round-Robin Rate Overview

Searching arXiv for the exact term and closely related round-robin rate concepts across domains.
“Rolling-Round-Robin Rate” is not a standard term in the arXiv literature. The closest paper-grounded interpretation is domain-dependent: in some works it denotes an effective per-user service rate induced by cyclic scheduling, in others an information transmission rate or capacity created by a round-robin mechanism, a per-round quota in packet schedulers, a dynamic time quantum that controls CPU rotation frequency, or a final secret key rate in round-robin differential phase-shift quantum key distribution. This suggests that the expression is best treated as an umbrella designation—an *Editor’s term*—for the rate-like quantity that emerges when service, access, assignment, or measurement proceeds in successive round-robin visits rather than through a single fixed-rate model [1802.00917][1701.08883][2108.09864][1103.3831][1505.08076].

## 1. Terminological status and scope

In the cited literature, no paper defines a metric literally called “Rolling-Round-Robin Rate.” Instead, each field instantiates a different nearest analogue. In wireless queueing, the closest quantity is the long-term per-user effective service rate under round robin, typically governed by a success probability times a scheduling fraction [1802.00917]. In covert-channel analysis, the relevant object is the achievable information rate in bits per time slot induced by a round-robin scheduler [1701.08883]. In hierarchical and deficit-based packet scheduling, the closest concept is a round-by-round quota or weighted service share that is rolled forward across rounds [1403.5712][2108.09864]. In CPU scheduling, the closest analogue is the time quantum, because it directly controls how frequently the scheduler rotates among ready processes [1605.00362]. In round-robin QKD, “rate” means secret key rate, sifted-key rate, gain, or clock rate, not scheduler throughput [1505.08076][2103.08452].

| Domain | Closest quantity | Representative expression |
|---|---|---|
| Small-cell wireless RR | Per-user effective service rate | $\mu^\Phi_{x_0}/K_{\mathrm{s}}$ [1802.00917] |
| Covert queueing channel | Capacity in bits per slot | $C=\max_{0\le p\le1}\frac{h(p)}{1+p}$ [1701.08883] |
| DRR ISP traffic control | Effective excess-bandwidth share over rounds | Quantum-proportional share via $Q_i$ [1403.5712] |
| HLS packet scheduling | Per-round quota-derived service | $F_i=\left\lfloor B_i/w_i^{\rm ac}\right\rfloor$ [2108.09864] |
| CPU RR variants | Rotation-controlling time quantum | $TQ=\sum BT_i/n$ or median-based $q_t$ [1605.00362][1103.3831] |
| RRDPS/RRDPTS QKD | Secret key rate / gain | $L\cdot R=2\bigl(Q(1-H(A|B))-e_{src}-(Q-e_{src})I_{AE}^U\bigr)$ for RRDPTS [2103.08452] |

A recurrent misconception is that round robin implies a single universal “rate.” The literature instead separates at least five different objects: access share, successful service rate, rotation frequency, information capacity, and secure key rate. Another misconception is that round robin is defined only by fairness of turn-taking. Several papers show that regularity of service opportunities, overlap structure, or quota propagation can matter as much as average share itself [1802.00917][2201.00881][2108.09864].

## 2. Wireless and queueing-theoretic service rates

In the downlink small-cell model of “Delay Analysis of Random Scheduling and Round Robin in Small Cell Networks” [1802.00917], each SAP serves $K_{\mathrm{s}}$ UEs, time is slotted, arrivals are Bernoulli with rate $\xi$, and under RR a typical UE is scheduled once every $K_{\mathrm{s}}$ slots. The paper’s closest exact analogue to a round-robin service rate is therefore the effective per-user service capability
\[
\frac{\mu^\Phi_{x_0}}{K_{\mathrm{s}}},
\]
where $\mu^\Phi_{x_0}$ is the conditional wireless service success probability. This quantity governs the stability condition
\[
\frac{\mu^\Phi_{x_0}}{K_{\mathrm{s}}}>\xi,
\]
the queue non-empty probability
\[
\tau_a^{\mathrm{RR}}=\min\left\{\frac{K_{\mathrm{s}}\xi}{\mu^\Phi_{x_0}},1\right\},
\]
and the mean delay
\[
\mathbf{D}_{x_0}^{\mathrm{RR}}=
\begin{cases}
\displaystyle
\frac{1-\frac{(K_{\mathrm{s}}+1)\xi}{2}}{\frac{\mu^\Phi_{x_0}}{K_{\mathrm{s}}}-\xi}-\frac{K_{\mathrm{s}}-1}{2}, & \text{if } \frac{\mu^\Phi_{x_0}}{K_{\mathrm{s}}}>\xi,\\[2ex]
+\infty, & \text{if } \frac{\mu^\Phi_{x_0}}{K_{\mathrm{s}}}\le \xi.
\end{cases}
\]
The paper further states that RR outperforms random scheduling in mean delay, with the gap
\[
\mathbf{D}_{x_0}^{\mathrm{RS}}-\mathbf{D}_{x_0}^{\mathrm{RR}}
=
\frac{K_{\mathrm{s}}-1}{2}\cdot
\frac{\mu^\Phi_{x_0}}{\mu^\Phi_{x_0}-K_{\mathrm{s}}\xi}>0,
\]
and emphasizes that the advantage is more pronounced under heavy traffic [1802.00917].

A distinct wireless interpretation appears in “Round-Robin is Provably Near-Optimal for Minimizing Age with HARQ over Heterogeneous Unreliable Multiaccess Channels” [2010.10861]. There, persistent round robin schedules users cyclically and persists with retransmissions until success. The cycle length is
\[
\eta=\sum_{m=1}^{N}\mathbb{E}[K_m],
\]
so each user’s long-run successful update frequency is effectively $1/\eta$. Under the two HARQ models studied, the paper derives asymptotic relative AoI gap bounds of about $6.4\%$ and $6.2\%$ for RR-P under the stated conditions, showing that in this setting the “rate” of successful cyclic service can be near-optimal even when users are heterogeneous [2010.10861].

## 3. Information, coding, and overlap-induced rates

In “A Covert Queueing Channel in Round Robin Schedulers” [1701.08883], the rate induced by round robin is not a service rate but an information transmission rate. The exact definition is
\[
R=\frac{\log M}{n},
\]
with units of bits per time slot. In the noiseless case the round-robin covert-channel capacity is
\[
C=\max_{0\le p\le1}\frac{h(p)}{1+p}\approx 0.6942 \text{ bits/time slot},
\]
and with packet drops of probability $\delta$ it becomes
\[
C=\max_{0\le p\le1}
\frac{h((1-\delta)p)-ph(\delta)}
{(1-p)+\delta p+2(1-\delta)p}.
\]
Here the round-robin mechanism itself creates the signaling alphabet, because Bob infers Alice’s bit from the pattern of his own service acknowledgments [1701.08883].

In “Round-Robin Streaming with Generations” [1206.3014], the closest analogue is effective throughput under cyclic generation service. One coded packet from each generation is sent sequentially and then the sender wraps around. The central analytic quantity is the expected delivery packet count
\[
E[T]=\sum_{t=0}^{\infty}(1-p_t),
\]
where $p_t=\Pr(T\le t)$ depends on how many round-robin service opportunities each generation has accumulated. The paper states that, because there is no feedback until the entire file has been downloaded, round robin may result in many superfluous transmissions for already decoded generations, but that the overhead per packet drops as generation size increases [1206.3014]. This is a throughput interpretation of a rolling RR rate: the file-level rate is determined by cyclic service opportunities and their induced completion distribution.

In “Balanced Nonadaptive Redundancy Scheduling” [2201.00881], round robin is a deterministic rolling assignment of each arriving job to $d$ consecutive servers modulo $n$. The long-run assignment frequency is perfectly balanced, expressed by
\[
\mathrm{LBF}_{\text{round-robin}}=1.
\]
The average overlap factor is
\[
\mathrm{AOF}_{\text{round-robin}}=\frac{n}{d^2},
\]
and the overlap diversity factor is
\[
\mathrm{ODF}_{\text{round-robin}}=\frac{3n}{2d^3+d}.
\]
The paper shows that this rolling assignment pattern can yield worse queueing performance than random despite perfect load balance, because the overlap geometry is unfavorable; the proposed block-design policy reduces average waiting time in the queue by up to $25\%$ compared to random and up to $100\%$ compared to round-robin [2201.00881]. This suggests that a “round-robin rate” defined only by average visit frequency can miss the more consequential structure of overlap classes.

## 4. Packet scheduling, bandwidth allocation, and rolling quotas

In ISP traffic control, “Deficit Round-Robin-Based ISP Traffic Control Scheme Enabling Excess Bandwidth Allocation in Shared Access Networks” [1403.5712] associates rate with the service that accumulates over repeated DRR visits. The scheme separates conformant and non-conformant traffic using token bucket meters, serves conformant traffic first, and uses DRR for non-conformant packets. The excess-capacity target is expressed through
\[
C_{ex}(t)=C-r_c(t),
\qquad
A(t)=\sum_{i=1}^{N}r_{nc,i}(t),
\]
and, when $A(t)>C_{ex}(t)$, the normalized fair rate $\alpha(t)$ satisfies
\[
C_{ex}(t)=\sum_{i=1}^{N}w_i\min(\alpha(t),r_{nc,i}(t)/w_i).
\]
The scheduler itself does not output a literal rolling rate variable. Rather, service emerges from deficit counters $DC_i$ and quanta $Q_i$, with the intended design that quanta are proportional to token generation rates. This suggests that the nearest “rolling round-robin rate” is the effective excess-bandwidth service a subscriber accumulates over successive DRR rounds, conditional on residual capacity after conformant traffic [1403.5712].

In hierarchical link sharing, “A Round-Robin Packet Scheduler for Hierarchical Max-Min Fairness” [2108.09864] makes the rate interpretation even more explicit. HLS computes a fair quota
\[
F_i=\left\lfloor\frac{B_i}{w_i^{\rm ac}}\right\rfloor,
\qquad
w_i^{\rm ac}=
\sum_{k\in{\rm child}(i)\cap(L_{\rm ac}\cup I_{\rm ac})}w_k,
\]
and an active child receives quota $w_iF_{p(i)}$. Unused quota is returned upward as residual and redistributed in surplus rounds. The global invariant is
\[
\sum_{i\in N}B_i+\sum_{i\in I\cup\{\rm root\}}R_i\equiv Q^*.
\]
Here rate is not set by a token bucket or measured moving average; it emerges from repeated rounds, quota propagation, and residual redistribution. The paper’s transmission-gap bound,
\[
\gamma^{\rm(HLS)}=\frac{2}{C}\left(\sum_{i\in I\cup L}w_i+\sum_{i\in L}L_i^{\max}\right),
\]
shows that the rolling quota process determines not only long-run allocation but also short-term revisit cadence [2108.09864].

## 5. CPU scheduling: quantum as round-robin rotation rate

In CPU scheduling papers, the nearest analogue to “Rolling-Round-Robin Rate” is the time quantum, because it determines how often the scheduler rotates among ready processes. “An optimized round robin cpu scheduling algorithm with dynamic time quantum” [1605.00362] states the tradeoff directly: a smaller quantum means a higher rotation rate but more context switches, while a larger quantum means a lower rotation rate and behavior closer to FCFS. Its DABRR algorithm sets
\[
TQ=\frac{TBT}{n}
=
\frac{\sum_{i=1}^{n}BT_i}{n},
\]
recomputed after each round on the current ready queue. Under the paper’s six benchmark cases, DABRR reports reductions relative to standard RR of about $41.23\%$ in waiting time and $30.70\%$ in turnaround time [1605.00362].

“A New Proposed Dynamic Quantum with Re-Adjusted Round Robin Scheduling Algorithm and Its Performance Analysis” [1103.3831] uses the median of current burst times as the quantum:
\[
q_t=
\begin{cases}
Y_{\frac{n+1}{2}}, & \text{if } n \text{ is odd},\\[4pt]
\dfrac{Y_{\frac{n}{2}}+Y_{\frac{n}{2}+1}}{2}, & \text{if } n \text{ is even}.
\end{cases}
\]
The quantum is recalculated after each cycle using the remaining burst times, and the queue is reordered in a smallest-largest alternating pattern. This is a direct example of a rolling RR rate in the sense of a round-dependent service slice [1103.3831].

The OMDRR family makes the rolling interpretation more literal. “An Optimum Multilevel Dynamic Round Robin Scheduling Algorithm” [1307.4167] describes a round-based update
\[
TQ_{r+1}=2TQ_r
\]
as reconstructed from the paper’s stepwise procedure, with a small-remainder rule that allows a process to finish if its remaining burst is less than $TQ/2$. “Simulation of an Optimum Multilevel Dynamic Round Robin Scheduling Algorithm” [1309.3096] describes an analogous multiplicative update
\[
TQ=TQ*F,
\]
together with a threshold condition based on $TQ/F$ and round-by-round warping of the time slice [1309.3096]. In both cases the “rate” is not throughput in the queuing sense but the amount of CPU granted per visit as the rounds progress.

A hybrid priority interpretation appears in “Characteristic specific prioritized dynamic average burst round robin scheduling for uniprocessor and multiprocessor environment” [1511.02498]. There the dynamic quantum is
\[
TQ=\left\lfloor\frac{TBT}{n}\right\rfloor,
\]
but the order of rotation is determined by a seven-feature priority score. The paper reports, in uniprocessor experiments, reductions relative to RR of $25.5\%$ in turnaround time and $35.2\%$ in waiting time, and in multiprocessor experiments reductions of $16.4\%$ and $35.6\%$, respectively [1511.02498]. At a more abstract level, “A Markov Chain Model for the Analysis of Round-Robin Scheduling Scheme” treats the nearest rate-like primitive as the forward rotation probability per quantum,
\[
p=P[X(n+1)=P_{i+1}\mid X(n)=P_i],
\]
with pure RR corresponding to $p=1$ [1006.2680].

## 6. Quantum key distribution: secret key rate as the round-robin rate

In QKD, round robin refers not to scheduler fairness but to random pairing of pulses in a packet. Consequently, the relevant rate is the secret key rate, or, more narrowly, the gain, sifted-key rate, and final secure throughput. “Experimental Passive Round-Robin Differential Phase-Shift Quantum Key Distribution” [1505.08076] reports a 500 MHz passive RRDPS system and states that it can generate secure key over 50 km with a bit error rate as high as 29%; the paper’s table includes a 53 km operating point with $e_b=0.312$ and estimated phase error $e_{ph}=0.00240$. Its asymptotic starting point is
\[
R\ge 1-H(e_b)-H(e_p),
\]
and its practical post-processing formula is
\[
K=N(1-H_{PA}-H_{EC}),
\qquad
H_{EC}=fH(e_b).
\]
Here “rate” is fundamentally secure-key yield under high error conditions, not a scheduling share [1505.08076].

“Experimental round-robin differential phase-shift quantum key distribution” [1505.08142] presents the active-delay RRDPS variant with 128 selectable delays and reports a final key rate of 15.54 bps at total loss 18 dB and $8.9\%$ error rate. The packet-level rate formula is
\[
R=Q\left[1-H(e_{\textrm{bit}})-H_{\mathrm{PA}}\right],
\]
where $Q$ is the average number of valid detections per $L$-pulse train. In this setting, “round-robin rate” can only mean key-generation rate, because the round-robin feature is Bob’s randomized delay choice among pulse pairs [1505.08142].

The most explicit rate expression appears in “Round-robin differential phase-time-shifting protocol for quantum key distribution: theory and experiment” [2103.08452]. For RRDPTS, the asymptotic secret key rate is
\[
L\cdot R
=
2\Bigl(Q(1-H(A|B)) - e_{src} - (Q-e_{src})I_{AE}^U\Bigr),
\]
with packet gain
\[
Q=\frac{1}{2}Q_x+\frac{1}{2}Q_z=\frac{1}{2}Q_x+\frac{1}{4}Q_{z,e}+\frac{1}{4}Q_{z,o}.
\]
The prefactor 2 reflects that each successful event yields two raw key bits. The paper reports that RRDPTS can achieve higher secret key rate than RRDPS in the condition of high quantum bit error rate, and gives proof-of-concept estimated SKRs of 1.5 kbit/s at 6 km, 1.0 kbit/s at 43 km, and 0.14 kbit/s at 80 km [2103.08452].

Across these QKD papers, a final misconception can be dispelled. The “round-robin” part of RRDPS or RRDPTS does not define a scheduler rotation rate at all; it defines a measurement structure over pulse pairs. The rate of interest is therefore cryptographic output rate. This contrasts sharply with wireless queueing, packet scheduling, and CPU scheduling, where the corresponding quantity is a service opportunity, visit frequency, or quota accumulation rate [1505.08076][1505.08142][2103.08452].

Taken together, the literature shows that “Rolling-Round-Robin Rate” has no single invariant meaning. In the strongest paper-grounded sense, it names the rate-like quantity produced by repeated cyclic access: $\mu^\Phi_{x_0}/K_{\mathrm{s}}$ in small-cell service, covert-channel capacity in bits per slot, quantum-controlled CPU rotation cadence, quota-derived packet service in DRR and HLS, or secure key rate in round-robin QKD. The exact formal object is therefore determined not by the phrase itself, but by which round-robin mechanism is rolling, what resource is being shared, and what notion of rate the underlying model treats as primary [1802.00917][1701.08883][2108.09864][2103.08452].

Source: https://www.emergentmind.com/topics/rolling-round-robin-rate