Queuing Theory to Guide the Implementation of a Heart Failure Inpatient Registry Program

Table 2

Two Simulated Solutions Using the HFR to Decrease Time to Connection Using a Queuing Theory Model

Post-HFR	Questions	λ	μ	S	ρ	P_S(S)	Ř_I	Ť_w
Scenario 2	What is the ideal HF patient arrival rate for optimal efficiency?	174^∗	218	1	0.8^†	0.8^∗	3.2^∗	13 h^∗
Scenario 3	What arrival rates can we accommodate if add 1 extra nurse?	348^∗	218	2^†	0.8^†	0.23^∗	0.15^∗	19 min^∗

HFR = heart failure readmission.
λ: average rate of patients identified as heart failure patients needing connections (per month).
μ: rate of connecting patients (average rate of patients being connected per month).
S: number of nurses.
ρ = λ/(Sμ): average utilization rate of the heart failure postdischarge connection program.
P_S(S) = ρ^S S p₀/S! (S-ρ) where p₀ = [1 + ρ/1! + ρ²/2! + … + ρ^(S-1)/(S-1)! + ρ^S/(1 + ρ/S)]⁻¹: probability of a busy period.
Ř_I = P_S(S) ρ/(S-ρ): average number of patients in line.
Ť_w: average patient waiting time.
↵∗ Numbers calculated are rounded for presentation clarity, answers in bold.
↵† Fixed variable.

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