Reference models¶

Models whose answer is known exactly, from a formula found in the textbooks of the subject, beside what the engine gives when it runs them. Every model here is one of the bundled models, so each line can be repeated with one command.

Case	Method	Model	Quantity	Exact	Naga	95% interval	Exact inside	Replications
M/M/1 queue	Discrete-event	`single_server`	mean number waiting	3.2	3.26	3.05 to 3.48	yes	20
M/M/2 queue (Erlang C)	Discrete-event	`two_teller_set`	mean number waiting	1.93	1.94	1.81 to 2.07	yes	20
M/G/1 queue (Pollaczek-Khinchine)	Discrete-event	`single_server_general`	mean number waiting	1.9	1.88	1.78 to 1.98	yes	20
Open Jackson network	Discrete-event	`job_shop_jackson`	mean time in the shop	8.38	8.37	7.95 to 8.79	yes	10
Logistic growth	System dynamics	`sd_population`	population at t = 25	600	600	no randomness	equal	1
Bass diffusion	System dynamics	`sd_bass_diffusion`	adopters at t = 10	835	835	no randomness	equal	1
SIR epidemic	System dynamics	`sd_sir`	peak number infected	404	404	no randomness	equal	1
Busy/idle cycle	Agent-based	`agents_server_utilization`	share of the time busy	0.444	0.447	0.442 to 0.451	yes	10
SIR epidemic of agents	Agent-based	`agents_sir_agents`	final number recovered	1,790	1,780	1,760 to 1,790	yes	20
Tank, average level	Rate-based	`tank_fill_and_drain`	time-average level	57	57	no randomness	equal	1
Tank, overflow	Rate-based	`tank_fill_and_drain`	quantity spilled	80	80	no randomness	equal	1

How to read it¶

Exact is the formula's value, computed from the parameters in the model file. Naga is the mean over the replications of a run made when this page was generated, and the interval is its 95% Student-t confidence interval over those replications. Numbers are printed at three significant digits; "inside" is decided before rounding.

A 95% interval misses the true value one time in twenty by chance. 6 lines here have an interval, so a "no" somewhere in the table is expected for about one seed in 4, with nothing wrong. Where a line has no interval, the model has no randomness and the two numbers are compared as printed.

The intervals are a few percent wide because the runs are kept to a few seconds each. An interval narrows with the square root of the simulated time: give a command below a longer --run-length or more --replications to see it close in on the exact value.

tools/gen_engine_docs.py makes the runs and writes this page; no number on it is typed. A test derives each value a second way (the Markov chain summed state by state, or the differential equation integrated numerically), repeats each run, and fails when the exact value is outside the interval or when a line here is not what the run gives.

The cases¶

M/M/1 queue¶

Single server (M/M/1) (single_server), mean number waiting: the KPI resource/Server/Queue Length Avg.

Exact: Lq = rho^2 / (1 - rho), rho = lambda / mu.

naga run single_server --replications 20 --run-length 10500 --warmup 500

M/M/2 queue (Erlang C)¶

Two tellers (resource set) (two_teller_set), mean number waiting: the KPI resource/Tellers/Queue Length Avg.

Exact: Lq = p0 a^c rho / (c! (1 - rho)^2), a = lambda / mu, rho = a / c, 1 / p0 = sum over n < c of a^n / n! + a^c / (c! (1 - rho)).

naga run two_teller_set --replications 20 --run-length 5500 --warmup 500

M/G/1 queue (Pollaczek-Khinchine)¶

Single server, general service times (M/G/1) (single_server_general), mean number waiting: the KPI resource/Server/Queue Length Avg.

Exact: Lq = lambda^2 E[S^2] / (2 (1 - rho)), rho = lambda E[S].

naga run single_server_general

Open Jackson network¶

Job shop (Jackson network) (job_shop_jackson), mean time in the shop: the KPI system/Avg Throughput Time.

Exact: W = sum of L_i / gamma: station i an M/M/c queue at the rate lambda_i of the traffic equations lambda_i = gamma_i + sum over j of lambda_j p_ji, L_i = Lq_i + lambda_i / mu_i.

naga run job_shop_jackson --replications 10 --run-length 6500 --warmup 500

Logistic growth¶

Population (logistic) (sd_population), population at t = 25: the KPI level/population_1/SD Population/Final.

Exact: P(t) = K / (1 + (K / P0 - 1) exp(-r t)).

naga run sd_population --run-length 25

Bass diffusion¶

Bass diffusion (sd_bass_diffusion), adopters at t = 10: the KPI level/bass_diffusion_1/SD Adopters/Final.

Exact: A(t) = m (1 - exp(-(p + q) t)) / (1 + (q / p) exp(-(p + q) t)).

naga run sd_bass_diffusion --run-length 10

SIR epidemic¶

SIR epidemic (sd_sir), peak number infected: the KPI level/sir_1/SD Infected/Max.

Exact: I_max = N - rho (1 + ln(S0 / rho)), rho = gamma N / beta.

naga run sd_sir

Busy/idle cycle¶

Server busy/idle cycle (agents) (agents_server_utilization), share of the time busy: the KPI agent/server_utilization_1/Servers/Busy/Avg Count.

Exact: lambda / (lambda + mu).

naga run agents_server_utilization --replications 10

SIR epidemic of agents¶

SIR epidemic (agents) (agents_sir_agents), final number recovered: the KPI agent/sir_agents_1/SIR People/R/Final.

Exact: the mean of N - S when the last infected agent recovers, summed over every path of the Markov chain: from (S, I) the next event is an infection with probability beta S / (beta S + gamma N), else a recovery.

naga run agents_sir_agents --replications 20 --run-length 300

Tank, average level¶

Tank: fill, overflow and run dry (tank_fill_and_drain), time-average level: the KPI level/Tank/Average.

Exact: the area under three straight lines (rise to full, full, fall to empty) over the run.

naga run tank_fill_and_drain

Tank, overflow¶

Tank: fill, overflow and run dry (tank_fill_and_drain), quantity spilled: the KPI level/Tank/Spilled.

Exact: (fill - draw) x the time the tank is full while the pump runs.

The same run as the line above.

What this does not show¶

Agreement on these cases says the engine's clock, queues, random streams, integrators and statistics are right where a formula can check them. It says nothing about a model that has no formula, which is most models: for those, see Model assertions and Experiments.