Hypercube queue of patrol units

Larson's hypercube model of patrol cars: calls arrive from each atom (beat, reporting area) at rate lam and go to the first free unit in the atom's order of preference; each unit serves at rate mu. The state is the set of busy units, \(2^n\) states for n units, and the stationary distribution is found exactly. From it come each unit's workload (the share of time busy), the share of calls each unit answers from each atom, the probability that every unit is busy (a lost call, or with queue = "wait" a call that waits; Larson's infinite-line model, the queue lumped into the all-busy state), the mean queue and wait, and with travel times the mean response travel time. Whatever the orders, the number of busy units follows Erlang's loss or delay system. The noisy-OR chance that patrols detect an event is nordet().

Usage

hypq(lam, pref, mu = 1, queue = "loss", tt = NULL)

Arguments

lam

The call rate of each atom: positive numbers.

pref

The dispatch orders: a matrix with one row per atom, each a permutation of the unit numbers 1 to n (at most 10 units), or a list of such vectors.

mu

The service rate of a unit (1 / mean service time): one positive number.

queue

"loss" (a call finding every unit busy is lost) or "wait" (it waits; the calls must be fewer than the units serve).

tt

NULL or the travel times: a matrix with one row per unit and one column per atom.

Value

An object of class hypq, a list: state, each set of busy units (busy, one character per unit, "1" busy) with its probability p; wl, the workloads; disp, the share of all calls answered at once by each unit (rows) from each atom (columns); queue; pfull, the probability that every unit is busy; lq and wq, the mean queue and wait (0 for "loss"); rho, the calls over what the units can serve; travel, NULL or the mean travel time overall (mean) and by atom (atom).

References

Larson, R. C. (1974). A hypercube queuing model for facility location and redistricting in urban emergency services. Computers and Operations Research, 1(1), 67-95. doi:10.1016/0305-0548(74)90076-8

Examples

pref <- rbind(c(1, 2, 3), c(2, 1, 3), c(3, 2, 1), c(3, 1, 2))
tt <- matrix(c(2, 5, 8, 6, 3, 7, 9, 6, 2, 7, 8, 3), 3)
hypq(c(0.4, 0.7, 0.3, 0.6), pref, mu = 1.5, tt = tt)
#> hypq: 3 units, rho 0.4444444; P(all busy) 0.109215
#> workloads: 0.3722186 0.4025169 0.4129779
#> mean travel time 3.944009