Let u be the transition rate from the lounge to the queue, let beta satisfy fracbetanu=eta, where eta>0 is constant, and let A=lfloor etamu/alpha rfloor. For each u, write bm pi^(nu)=(pi_q,l^(nu))_q,l for the stationary distribution of the system with lounge functionality. Stationary-distribution approximation conjecture. As u→0 with beta/nu=eta, sup |bm pi^(nu)-bm pi^(0)|→0. Thus, the system with positive transition rate u is approximated by the system with u=0 and A=lfloor etamu/alpha rfloor. The conjecture formalizes the observed improvement of the approximation as u decreases in the regime where lounge waiting times become very long while the comfort parameter beta decreases proportionally. No resolution or proof is supplied in the source.

Stationary-distribution approximation for lounge-from-queue systems

Let $u$ be the transition rate from the lounge to the queue, let $\beta$ satisfy

\frac{\beta}{\nu}=\eta,

where $\eta>0$ is constant, and let $A=\lfloor\eta\mu/\alpha\rfloor$ . For each $u$ , write ${\bm \pi}^{(\nu)}=(\pi_{q,l}^{(\nu)})_{q,l}$ for the stationary distribution of the system with lounge functionality. Stationary-distribution approximation conjecture. As $u\to0$ with $\beta/\nu=\eta$ ,

\sup \left\|{\bm \pi}^{(\nu)}-{\bm \pi}^{(0)}\right\|\to0.

Thus, the system with positive transition rate $u$ is approximated by the system with $u=0$ and $A=\lfloor\eta\mu/\alpha\rfloor$ . The conjecture formalizes the observed improvement of the approximation as $u$ decreases in the regime where lounge waiting \times become very long while the comfort parameter $\beta$ decreases proportionally. No resolution or proof is supplied in the source.

Progress summary

Solved

An unverified posted attempt claims a complete proof, with stronger total-variation convergence and a linear error rate, but the conjecture is not independently confirmed.

The conjecture asserts that, under proportional scaling of the lounge-transition and comfort rates, the stationary law of the positive-rate system converges to that of the zero-rate approximation. It appears in work by Riya Sultana, Khushboo Agarwal, and Veeraruna Kavitha, first posted April 28, 2025.

Posted attempt

A reader-written argument claims a complete proof: the total population is identified with an $M/M/1$ chain, tightness and limiting balance equations yield convergence, and a regenerative coupling allegedly gives total-variation distance $O(\nu)$ . The attempt has not been independently verified.

Current status (as of August 2026): The conjecture has a claimed but unverified proof; no independent verification is recorded, so the mathematical question remains unsettled.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Riya Sultana, Khushboo Agarwal and Veeraruna Kavitha, “Queue or lounge: strategic design for strategic customer”, arXiv:2504.19889 (2025).

1 week ago

Proof, strengthened to total-variation convergence with an explicit linear rate.

Fix $0<\lambda<\mu$ , put $\rho=\lambda/\mu$ , $\delta=\mu-\lambda$ , and impose the source's scaling $\beta=\eta\nu$ . Its thresholds are

A=\left\lfloor\frac{\eta\mu}{\alpha}\right\rfloor, \qquad B_\nu=\left\lceil\frac{\delta}{\nu}-A\right\rceil \longrightarrow\infty.

Write $Q_t,L_t$ for the queue and lounge populations, and extend all stationary laws by zero to the common state space

\mathsf S=\{(0,0)\}\cup\{(q,l):q\ge1,\ l\ge0\}.

The source's instantaneous-transfer assumption ensures that the server is busy whenever $N_t=Q_t+L_t>0$ . Therefore, for every $\nu\ge0$ , the total population $N_t$ is exactly the same $M/M/1$ birth–death chain, with birth rate $\lambda$ and death rate $\mu$ . In particular,

\pi^{(\nu)}\{Q+L=j\}=(1-\rho)\rho^j \qquad(j\ge0,\ \nu\ge0). \tag{1}

All the stationary laws are consequently uniformly geometrically tight. On every fixed finite subset of $\mathsf S$ , eventually $l<B_\nu$ , and the generators converge because the only remaining differing transition has rate $\nu l\to0$ . Every subsequential limit therefore satisfies the limiting stationary balance equations. Regeneration at $(0,0)$ makes the limiting invariant probability unique, and (1) preserves all mass. Hence

\|\pi^{(\nu)}-\pi^{(0)}\|_{\rm TV}\longrightarrow0.

A regenerative coupling gives a stronger quantitative conclusion. Use the same arrival and service clocks in both systems, so their total-population trajectories coincide. For one empty-to-empty cycle, let

I\sim\operatorname{Exp}(\lambda),\qquad T=\text{busy-period length},\qquad C=I+T,\qquad J=\int_0^T N_s\,ds,\qquad M=\sup_{s\le T}N_s.

Before the two trajectories separate, a spontaneous transfer has rate $\nu L_s\le\nu N_s$ ; otherwise routing differs only when $M\ge B_\nu$ . Therefore the regenerative occupation formula and Cauchy–Schwarz give

d_{\rm TV}(\pi^{(\nu)},\pi^{(0)}) \le \frac{\nu\,\mathbb E[CJ] +(\mathbb EC^2)^{1/2} \mathbb P(M\ge B_\nu)^{1/2}} {\mathbb EC}. \tag{2}

Gambler's ruin for the busy excursion gives

\mathbb P(M\ge B) =\frac{(1-\rho)\rho^{B-1}}{1-\rho^B} \le\rho^{B-1}.

The killed birth–death equations yield explicitly

\mathbb EC=\frac1\lambda+\frac1\delta,\qquad \mathbb EC^2 =\frac2{\lambda^2} +\frac2{\lambda\delta} +\frac{2\mu}{\delta^3},

and

\mathbb E[CJ] =\frac{\mu}{\lambda\delta^2} +\frac{2\mu(\lambda+\mu)}{\delta^4}.

Substitution into (2) proves

\boxed{ d_{\rm TV}(\pi^{(\nu)},\pi^{(0)}) \le \frac{ \nu\!\left( \frac{\mu}{\lambda\delta^2} +\frac{2\mu(\lambda+\mu)}{\delta^4} \right) + \left( \frac2{\lambda^2} +\frac2{\lambda\delta} +\frac{2\mu}{\delta^3} \right)^{1/2} \rho^{(B_\nu-1)/2} }{ \frac1\lambda+\frac1\delta } =O(\nu). }

Since the conjectured coordinatewise supremum is bounded by total variation, this proves the original claim and gives a strictly stronger norm and convergence rate.

Source: Sultana, Agarwal, and Kavitha, Queue or lounge: strategic design for strategic customer, Conjecture 1 and equations (7)–(8), https://arxiv.org/abs/2504.19889 .

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