Abstract

This study develops a stochastic queueing framework to model the dynamical interaction between proliferating tumor cells and the adaptive immune system, incorporating the critical phenomenon of immune checkpoint inhibition. The immune response is conceptualized as a multi-server service system where CD8+ T cells and natural killer cells act as parallel servers eliminating cancerous cells, while tumor cell division follows a controlled birth process with logistic growth constraints. The novelty lies in embedding a state-dependent immune evasion mechanism through a dimensionless checkpoint parameter κ∈[0,1] that modulates the effective killing rate as a function of tumor burden via the saturation function ϕ(n)=n/(K+n). The resulting process {N(t),t≥0} constitutes a non-homogeneous birth–death Markov chain with the following transition rates: λn=λ1−nK1{n<K},μn=nμ(1−κϕ(n)),1≤n<c,cμ(1−κϕ(n)),n≥c, We derive the exact stationary distribution in closed form, establishing the fundamental stability condition ρeff=λ/[cμ(1−κ)]<1 for tumor elimination. The expected tumor burden L and mean elimination time W are computed explicitly. A critical threshold κc=1−λ/(cμ) emerges, above which the immune system loses control regardless of other parameters. Heavy-traffic analysis reveals quadratic divergence L∼K/(1−ρeff)2 as ρeff→1−, explaining catastrophic tumor escape. Extensive Monte Carlo simulations (n=104 replications) validate theoretical predictions with relative errors <5% and p-values >0.05 from two-sample t-tests. The model provides quantitative tools for optimizing checkpoint inhibitor dosages and predicting patient-specific responses in immuno-oncology.

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