Package jline.api.qsys

Enum GeoGeo1Convention

java.lang.Object
java.lang.Enum<GeoGeo1Convention>
jline.api.qsys.GeoGeo1Convention
All Implemented Interfaces:
Serializable, Comparable<GeoGeo1Convention>, java.lang.constant.Constable
public enum GeoGeo1Convention extends Enum<GeoGeo1Convention>
Observation epoch for a discrete-time Geo/Geo/1 queue.

The two entries are not two different systems. They are the same system -- Daduna's LA-rule (all arrivals and departures occur at the end of their slot) combined with the D/A-rule (when both fall on one epoch, the departure is resolved first) -- observed at two different instants within the slot boundary. The queue-length process is X(t+1) = X(t) - D(t) + A(t); LAS_DA is the law of X, taken after both events, and EAS is the law of the intermediate Y(t) = X(t) - D(t), taken after the departure and before the arrival.

Since the two epochs are one departure apart and the departure rate equals the arrival rate a in steady state, the means differ by exactly a in population and, by Little's law, by exactly one slot in sojourn time. The queueing delay is the same under both.

Note that the regulation rule itself is not what separates them. Daduna (2001), section 3.1, records that with ample waiting room the D/A-rule and the A/D-rule yield the same stationary boundary distribution, because a service occupying at least one whole slot prevents a job from departing in its own arrival slot. That is also why the LDES engine's intra-slot event ordering does not move the estimates; it fixes a convention that would otherwise be emergent rather than modelled.

Why two constants suffice. The literature also names a late-arrival system with immediate access (LAS-IA), which raises the question of whether a third constant is missing. It is not, because the access rule and the observation epoch are not separately identifiable. Delaying by one slot the first slot in which an arriving job may be served shifts every mean the same way that stepping the observation epoch back by one does: verified by slot-level simulation, the delayed-access system's mid-slot population equals the immediate-access system's boundary population (0.69897 against 0.69897 at a=0.3, s=0.6; 0.53272 against 0.53338 at a=0.2, s=0.5, within sampling noise). So the reachable means form the one-parameter family E[T] = (1-a)/(s-a) + m, E[N] = a E[T], indexed by an integer slot offset m, and LAS-IA and EAS name the same member of it. The two constants below are m = 0 and m = -1, the only two that carry standard names; any further offset re-indexes the same family rather than describing a new system.

Reference: H. Daduna, Queueing Networks with Discrete Time Scale, LNCS 2046, Springer 2001, section 2.1 (figures 2.2 and 2.3, theorem 2.3, corollary 2.7).

  • Enum Constant Details

    • LAS_DA

      public static final GeoGeo1Convention LAS_DA
      Late arrival system with delayed access: the state at the slot boundary, after both the departure and the arrival of the slot. A job never departs in its own arrival slot and the service time is geometric on {1,2,...} with mean 1/s.

      This is the convention realized by the LDES engine when a Source with a Geometric(a) interarrival time feeds an FCFS Queue with a Geometric(s) service time, because both are supported on {1,2,...}.

      Matches Daduna corollary 2.7 term for term, including the tail ratio bq/(cp) in his notation (b = a, p = s).

    • EAS

      public static final GeoGeo1Convention EAS
      Early arrival system: the intermediate state, after the departure of the slot and before its arrival. Equivalently, the sojourn is measured net of the arrival slot, so the effective service time is geometric on {0,1,...} with mean (1-s)/s.

      This is the process observed at t+1/2 in Daduna's figure 2.3. Its empty probability is 1-r, NOT 1-rho: a job can arrive and be cleared within one slot, so the empty-at-epoch probability and the busy-slot fraction come apart even though the utilization is still rho in both.

  • Method Details

    • values

      public static GeoGeo1Convention[] values()
      Returns an array containing the constants of this enum type, in the order they are declared.
      Returns:
      an array containing the constants of this enum type, in the order they are declared

    • valueOf

      public static GeoGeo1Convention valueOf(String name)
      Returns the enum constant of this type with the specified name. The string must match exactly an identifier used to declare an enum constant in this type. (Extraneous whitespace characters are not permitted.)
      Parameters:
      name - the name of the enum constant to be returned.
      Returns:
      the enum constant with the specified name
      Throws:
      IllegalArgumentException - if this enum type has no constant with the specified name
      NullPointerException - if the argument is null