Positive recurrence of the fork-join LLM engine network

Consider a fork-join network of four LLM agents indexed by $j\in\{0,1,2,3\}$. Requests arrive at agent $0$ with rate $\lambda$, fork into tasks processed by agents $1$ and $2$, and are merged and routed to agent $3$ after both tasks complete. Let the token sizes at agent $j$ be $(m^j_p,m^j_d)$, and let agent $j$ have parameters $b^j$ and $t^j_b$. Fork-join stability conjecture. If

$$\lambda(m^j_p+m^j_d)<b^j/t^j_b\quad\text{for each }j\in\{0,1,2,3\},$$

then, under any $(K^j_p,K^j_d)$-FCFS work-conserving scheduling algorithm at each agent $j\in\{0,1,2,3\}$, the DTMC describing the system is positive recurrent. This proposes stability of the fork-join network whenever the offered load is below the service capacity at every agent; the source does not provide a resolution of the claim.