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6.3 observable data of d3t systems

6.3.2 Load observables

i=1

yTi(τ),

where]denotes multiset addition.

Hence, the total output trajectory aggregates the outputs of the dispatcher and all load sources and transporters.

Observer instances preprocess the total output trajectory yM of a D3T in-stance M. They provide recordable simulation data. Specifically, each ob-server model implements a combination of

1. scalar values,

2. event trajectories, and 3. state trajectories,

as simulation output. While a scalar value aggregates a quantity over the whole simulation run, a trajectory provides online time-discrete output (event trajectories), or time-continuous output (state trajectories).

6.3.2 Load observables

overview Load observables relate the transport dynamics to individual service quality. Figure6.9, Table6.10and Table6.11summarize the observ-ables regarding the arrival and transport of loads.

6.3 observable data of d3t systems 59

Table 6.10:Load observables in D3T models.

Symbol Domain / Value Observable

l ∈L=N load index

#l ∈M origin

l ∈M destination

dl =d(#l,l) direct travel time tl ∈R+0,tl>tl−1 arrival epoch

∆tl =tl−tl−1,∆t1 =t1 interarrival time

tXl >tl assignment epoch

tpl >tXl pick-up epoch

tdl >tpl+dl delivery epoch, departure epoch ql =tXl −tl queueing time

wl =tpl−tl waiting time el =tpl−tXl lead time xl =tdl −tXl service time πl =tdl −tpl travel time

sl =tdl −tl sojourn time, system time ˆ

ql =ql/dl relative queueing time ˆ

wl =wl/dl relative waiting time ˆ

el =el/dl relative lead time ˆ

xl =xl/dl relative service time ˆ

πll/dl relative travel time ˆ

sl =sl/dl relative sojourn time

nl =l(tl) system size at arrival epoch

Table 6.11: Derived load observables of D3T models.

Symbol Domain / Value Observable

L0(t) ={l:tl6t} arrival set

Ld(t) =

l:tdl 6t departure set

L(t) =

l:tl6t < tdl =L0(t)\ Ld(t) system set

n0L(t) =|L0(t)| arrival number

ndL(t) =|Ld(t)| departure number

l(t) =|L(t)| system size

nl =l(tl) system size at arrival epoch

lk(t) = 1t Rt

0I{t:l(˜˜ t)=k}dt0 system size fraction

nk(t) =|{l:tl6t,nl=k}| arrival system size frequency

60 modelling and simulating d3t

load epochs The arrival timetlof thel-th load is implicitly defined as1 (?,(l,σ,#,))∈yM(tl,c).

Similarly, the origin#l= #and destinationl =. Define thedirect travel timeas the metric distance from origin to destination:

dl=d(#l,l). Theinter-arrival timeis

∆tl=tl−tl−1,

with∆t1 =t1. The dispatcher assigns the load for transport to a transporter iat theassignment epochtXl . Its implicit definition is

(X,(l,il))∈yM(tXl ,c),

where il is the index of the transporter the load l is assigned to. Similarly, the implicit definitions of the pick-up epochtpl and the delivery epochtdl are

(p,(i,P))∈yM(tpl,c),l∈P, (d,(i,D))∈yM(tdl,c),l∈D.

load times In queueing theory, the assignment epoch is the time instant at which transporter i starts servicing loadl. Hence, we call the period be-tween arrival and assignment

ql=tXl −tl

the queueing time. The queueing time is less than the waiting time between arrival and pick-up

wl=tpl−tl,

which also includes thelead timebetween assignment and pick-up el=tpl−tXl .

During the lead time, a transporter travels to the origin of the load. In queue-ing theory, the transporter would already serve the load. However, in trans-portation, the transporter has not picked up the load yet. Besides the lead time, theservice timebetween assignment and delivery

xl=tdl−tXl

1 A note on potential confusion of queueing and physical notions here: in queueing theory, arrivalof a job or customer refers to the point in time a jobentersthe system, i.e. is known to and dealt with by the queueing system. In transport, arrival refers to the physical arrival of a load at its destination. In practice, we will not speak of arrival of a load, but rather of the deliveryof a load – which entails that the loaddeparts(or exits, leaves) the system. Hence, if not otherwise stated, arrival of a load means the arrival of a load to the (queueing) system.

Similarly, departure of a load refers to the load being delivered and subsequently, leaving the queueing system.

6.3 observable data of d3t systems 61 as defined in queueing theory also comprises thetravel timebetween pick-up and delivery

πl=tdl−tpl,

which must be at least the direct travel time, πl>dl.

Equality holds if the transporter travels directly from the load origin to the load destination. To conclude, the overallsojourn timeorsystem timeof load lbetween arrival and delivery

sl=tdl−tl adds up as

sl=wll=ql+xl=ql+ell.

For each of these timesql,wl,el,xll,sl, we also consider the correspond-ingrelativeperiod normalized by the direct travel timedl:

ˆ ql= ql

dl, ˆwl= wl

dl, ˆel= el

dl, ˆxl= xl

dl, ˆπl= πl

dl, ˆsl= sl dl. load sets Thearrival set

L0(t) ={l:tl6t}

is the set of all loads that have arrived by timet. Similarly, the departure set Ld(t) =

l:tdl 6t

is the set of all loads that have departured by timet. Clearly,Ld(t)⊂L0(t). Thesystem set

L(t) =

l:tl6t < tdl

=L0(t)\ Ld(t) is the set of loads currently in the system at timet. load numbers Thearrival number

n0L(t) =|L0(t)|

is the number of loads that have arrived by timet. Similarly, the departure number

ndL(t) =|Ld(t)|

is the number of loads that have departured by timet. Thesystem size l(t) =|L(t)|=n0L(t) −ndL(t)

62 modelling and simulating d3t

is the number of loads currently in the system. The system size at arrival epochstl, excluding the arrivals attl, is denoted

nl=l(tl).

The fraction of time the system size equalskup to time t(system size frac-tion) is

lk(t) = 1 t

Zt

0

I{t:l(˜˜ t)=k}dt0.

The number of arrivals to a system of sizekup to timet(arrival system size frequency) is

nk(t) =|{l:tl6t,nl=k}|.

system size busy periods and return time Given an integer number c∈N, what is the distribution of periods during which the D3T system has more than c loads? We call these periods c-busy periods. The intermediate periods are the c-idle periods. The start epochs of the c-busy periods up to timetare

tb,c(t) =˜t:t˜6t,l(˜t)>c,l(t˜)< c ,

and the start epochs of the c-idle periods up to timetare ti,c(t) ={0}∪t˜:t˜ 6t,l(˜t)< c,l(˜t)>c .

The start epochs of the c-idle periods are the end epochs of the c-busy pe-riods, and vice versa. The symbol l(˜t) refers to the right limit limt%˜tl(t).

The time-ordered sequences are(tb,cn )nand(ti,cn)nwith06tb,c1 6tb,c2 6. . . and0=ti,c0 6ti,c1 6ti,c2 6. . .. Observe that by definition

0=ti,c0 6tb,c1 6ti,c1 6tb,c2 6ti,c2 6. . .

Thesystem size c-busy periodsandsystem size c-idle periodsare bcn=ti,cn −tb,cn

icn=tb,cn −ti,cn−1.

Thesystem size return timesare the system sizeN-busy times rn=bNn,

where N is the number of transporters. The number of arrivals during a system size c-busy period up to timetis

nb,c(t) ={l:tl6t,nl>c},

(system size c-busy arrival number) and the number of arrivals during a system size c-idle period up to timetis

ni,c(t) ={l:tl6t,nl< c},

6.3 observable data of d3t systems 63

thesystem size c-idle arrival number. Obviously, n0L(t) =nb,c(t) +ni,c(t).

Thesystem size c-busy arrival fractionof arrivals up to timetis pb,c(t) = nb,c(t)

n0L(t) .

The system size delay fractionis the system size N-busy arrival fraction up to timet,

pb(t) =pb,N(t),

whereNis again the number of transporters.

payload-weighted travel time The payload-weighted travel time of loadlis

ψl= Ztd

l

tpl

(nil(t))−1dt,

where nil(t) is the payload size trajectory of the transporter il that load l was assigned to.