Analytic queueing model for ambulance services

We present predictive tools to calculate the number of ambulances needed according to demand of entrance calls and time of service. Our analysis discriminates between emergency and non-urgent calls. First, we consider the nonstationary regime where we apply previous results of first-passage time of one dimensional random walks. Then, we reconsider the stationary regime with a detailed discussion of the conditional probabilities and we discuss the key performance indicators.

💡 Research Summary

The paper develops an analytical queueing framework to determine how many ambulances are required for an Emergency Medical Service (EMS) based on the average inter‑call interval and the average service time. The system is modeled as an M‑server birth‑death process (M/M/s/∞/∞) with Poisson arrivals at rate λ = 1/T_C and exponential service times with rate μ = 1/T_S. The state variable n denotes the total number of calls currently in the system (both being served and waiting). When n = 0 all ambulances are idle; for 0 < n < M there are n ambulances busy and no queue; n = M is the critical state where every ambulance is occupied; and n > M indicates a saturated system with a queue of length n − M.

Non‑stationary analysis – The authors focus on the time required for the system to reach the critical state for the first time. By treating the occupation number as a one‑dimensional random walk with a reflecting boundary at 0 and an absorbing boundary at M, they apply known results for the mean first‑passage time (MFPT). Transition rates are ω⁺_n = λ (independent of n) and ω⁻_n = n μ for n ≤ M, ω⁻_n = M μ for n ≥ M. The MFPT from an initial state n is expressed in closed form (Eq. 3) using the dimensionless ratio γ = μ/λ = T_C/T_S. Averaging over all possible initial states yields ⟨T⟩, a function of λ, μ, and M. Numerical illustrations (Fig. 2) show that ⟨T⟩ grows non‑linearly with the mean inter‑call time T_C; for realistic service times (T_S ≈ 50 min) and fleets of 5–9 ambulances, the system remains below the critical state throughout the typical eight‑hour high‑call‑volume period provided T_C exceeds roughly 13–16 min, depending on fleet size. This result offers a quick, real‑time tool for staffing decisions.

Stationary analysis – Introducing the traffic intensity ρ = λ/(M μ), the authors derive the steady‑state probabilities π_n for all n. When ρ < 1 a normalizable distribution exists: for 0 ≤ n ≤ M, π_n ∝ (M! / n!) ρⁿ; for n ≥ M, π_n ∝ ρⁿ. The normalizing constant S is given in Eq. (7). The probability that all servers are busy (full occupation) is P(occup) = ∑_{n≥M}π_n = (M ρ)ᴹ/(M! (1 − ρ) S).

Conditional on full occupation, the number of waiting calls follows a geometric distribution P(k | occup) = ρᵏ (1 − ρ), k ≥ 0. Consequently, the mean queue length is ⟨L⟩ = ρ/(1 − ρ) and its standard deviation σ_L = √