Signature
rho = lambda / mu; W_q = rho / (mu - lambda)
| Inputs | Definition | Unit |
|---|---|---|
lambda | Poisson arrival rate at the server | arrivals per unit of time, for example patients per hour |
mu | Rate at which the server completes patients when busy, the reciprocal of the mean service time | patients per unit of time |
rho | Arrival rate divided by service rate, the long-run proportion of time the server is busy | proportion, below 1 for a stable queue |
|---|---|---|
W_q | Expected time from arrival to the start of service, excluding the service itself | time, in the reciprocal of the unit of the rates, for example hours |
Function
Arrival process count, gap and arrival-time function
Maps an arrival rate, constant or varying with time, to the number of patients or other entities that arrive in a window, the gaps between successive arrivals and the clock time of each arrival. The three descriptions carry the same information: S_n is the time of the nth arrival, X_i the gap before arrival i and N(t) the number of arrivals up to time t, in the notation of the Arrival Process article. The records cover Poisson counts, time-varying rates, sampling of gaps and arrival times in a discrete event simulation, and the single-server queueing result that links the arrival rate to waiting.
Try this function
Implementations
Excel
Single-server expected wait in one cell
With named cells ArrivalRate and ServiceRate in the same time unit, the formula returns the expected wait before service. It is meaningful only when ArrivalRate is below ServiceRate.
=(ArrivalRate/ServiceRate)/(ServiceRate-ArrivalRate)
Assumptions
Steady state with arrivals below service capacity
The arrival rate is below the service rate and both have been constant long enough for the queue to settle. With lambda at or above mu there is no steady state and the queue grows without limit.
Poisson arrivals, exponential service and one server
Arrivals form a homogeneous Poisson process, service times are exponential and independent, and one server works through the queue. Services with several servers, such as a ward of beds, need multi-server results or a simulation.
Worked examples
Single-server wait at 80% utilisation
An illustrative clinic receives 2 patients an hour and its single clinician can see 2.5 an hour. Utilisation is 0.8 and the expected wait before service is 1.6 hours.
lambda = 2; mu = 2.5; rho = 0.8; W_q = 1.6
Single-server wait at 90% utilisation
A rise in arrivals of one eighth, to 2.25 an hour, takes utilisation to 0.9 and the expected wait to 3.6 hours, more than double, which is the steep rise near full utilisation described in the article.
lambda = 2.25; mu = 2.5; rho = 0.9; W_q = 3.6
Common errors
Applying the single-server wait when arrivals exceed capacity
With 3 arrivals an hour and a service rate of 2.5, the formula gives minus 2.4 hours. The queue has no steady state and grows through the session, so the result has to come from a time-dependent model or simulation.
Reporting the wait before service as the time in the system
W_q excludes the service itself. In the 80% example the expected wait is 1.6 hours, while the expected time from arrival to leaving is 2 hours, the wait plus the mean service time of 0.4 hours.
Sources
Single-server queue with Poisson arrivals and exponential service
Gallager RG. Discrete Stochastic Processes, course text for MIT 6.262, chapter 6 (Markov processes with countable state spaces). MIT OpenCourseWare; 2011. Section 6.5 on birth-death processes and the M/M/1 queue, equations 6.39 to 6.42: utilisation rho equal to lambda over mu, required to be below 1, and expected queueing time 1/(mu minus lambda) minus 1/mu, equal to rho/(mu minus lambda).
Wait times as outcomes of constrained-resource simulation models
Karnon J, Stahl J, Brennan A, Caro JJ, Mar J, Möller J. Modeling using discrete event simulation: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-4. Value in Health. 2012;15(6):821-827. Section on constrained resources: common outcomes include flow times, wait times, throughput and resource utilisation, and constrained resources should be modelled when increased referral rates lengthen waits.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0