Le eb e, Ma io
A icle
A con olled disc e e- ime queueing sys em as a model o
he o de s o wo compe ing companies
Games
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Sugges ed Ci a ion: Le eb e, Ma io (2024) : A con olled disc e e- ime queueing sys em as a model
o he o de s o wo compe ing companies, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 3,
pp. 1-8,
h ps://doi.o g/10.3390/g15030019
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Ci a ion: Le eb e, M. A Con olled
Disc e e-Time Queueing Sys em as a
Model o he O de s o Two
Compe ing Companies. Games 2024,
15, 19. h ps://doi.o g/
10.3390/g15030019
Academic Edi o : Ul ich Be ge
Recei ed: 30 Ap il 2024
Re ised: 26 May 2024
Accep ed: 27 May 2024
Published: 29 May 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
games
A icle
A Con olled Disc e e-Time Queueing Sys em as a Model o he
O de s o Two Compe ing Companies
Ma io Le eb e
Depa men o Ma hema ics and Indus ial Enginee ing, Poly echnique Mon éal, C.P. 6079,
Succu sale Cen e- ille, Mon éal, QC H3C 3A7, Canada; mle eb [email p o ec ed]
Abs ac : We conside wo companies ha a e compe ing o o de s. Le
X1(n)
deno e he numbe
o o de s p ocessed by he i s company a ime
n
, and le
τ(k)
be he i s ime ha
X1(n)<j
o
X1(n) =
, gi en ha
X1(
0
) = k
. We assume ha
{X1(n)
,
n=
0, 1,
. . .}
is a con olled disc e e- ime
queueing sys em. Each company is using some con ol o inc ease i s sha e o o de s. The aim o he
i s company is o maximize he expec ed alue o
τ(k)
, while i s compe i o ies o minimize his
expec ed alue. The op imal solu ion is ob ained by making use o dynamic p og amming. Pa icula
p oblems a e sol ed explici ly.
Keywo ds: dynamic p og amming; di e ence equa ions; linea equa ions; i s -passage ime; homing
p oblem
AMS Subjec Classi ica ion: P ima y 93E20; Seconda y 60K25
1. In oduc ion
Many pape s ha e been published on he op imal con ol o queueing sys ems. O en
he au ho s assume ha i is possible o con ol he se ice a e; see, o example, Laxmi
e al. [
1
], Chen e al. [
2
] and Tian e al. [
3
]. In o he pape s, he aim is o con ol he a i al
o cus ome s in o he sys em; see Wu e al. [
4
]. Some imes, i is assumed ha he e a e
di e en ypes o cus ome s; see Wen e al. [
5
] and Dudin e al. [
6
]. Ano he p oblem
ha is o en conside ed is o de e mine he op imal numbe o se e s in he sys em; see
Asadzadeh e al. [7].
In his pape , we conside a single-se e queueing model in disc e e ime. O de s
a i e one a a ime. We assume ha he ime needed o an o de o a i e is a andom
a iable A ha ollows a geome ic dis ibu ion wi h pa ame e 2 pA, so ha
P[A=n]=(1−2pA)n−12pA o n=1, 2, . . . (1)
Mo eo e , he se ice ime is a andom a iable
S
ha ing a geome ic dis ibu ion
wi h pa ame e pS.
The e a e wo compe ing companies. In equilib ium, o de s a i e a each company
acco ding o a andom a iable
Ai
ha ing a geome ic dis ibu ion wi h pa ame e
pA
, o
i=1, 2.
Ins ead o sol ing an op imal con ol p oblem, ou aim in his pape is o p esen a
s ochas ic dynamic game ha can se e as a model o he beha io o companies com-
pe ing o o de s. We mus , o cou se, make simpli ying assump ions in o de o ob ain a
ma hema ically ac able p oblem. Howe e , we belie e ha he model is ealis ic enough
o be use ul.
Le
X1(n)
be he numbe o o de s being p ocessed by he i s company a ime
n
,
and le
τ(k) [= τ(k,j, )] :=in {n≥0 : X1(n)<jo X1(n) = |X1(0) = k}, (2)
Games 2024,15, 19. h ps://doi.o g/10.3390/g15030019 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 19 2 o 8
whe e j≤k≤ and j,k, ∈N.
Now, suppose ha each company can y o use some o m o con ol in o de o
inc ease i s sha e o he o de s. Mo e p ecisely, we assume ha he i s company uses he
con ol
un∈ {
0,
pA}
a ime
n
o inc ease he a i al a e o i s o de s. Simila ly, he second
company uses n∈ {−pA, 0}a ime n o dec ease he o de a i al a e o i s compe i o .
Rema k 1. W i ing ha
n=−pA
means ha Company 2 is making some e o s ( o ins ance,
by lowe ing i s p ices) in o de o educe o ze o he a e a which Company 1 ecei es o de s, unless
his company also ies o inc ease o a leas keep i s sha e o he o de s.
Le
pA(n):=pA+un+ n o n=1, 2, . . . (3)
We de ine he cos unc ion
J(k) =
τ(k)−1
∑
n=0nq1u2
n+q2 2
n+λ( n−un)o, (4)
whe e
q1
,
q2
and
λ
a e posi i e cons an s. We look o he alues
u∗
n
and
∗
n
o he con ol
a iables ha a e such ha he expec ed alue o J(k)is minimized.
The idea behind he cos unc ion is as ollows: since he pa ame e
λ
is posi i e, i
is
la ge, hen Company 1 wan s o maximize he ime i s ays in business. To do his, i would
like o use he con ol
un=pA
; howe e , his leads o quad a ic con ol cos s. Company 2,
o i s pa , would like o use
n=−pA
in o de o bank up Company 1 as quickly as
possible, bu his also en ails quad a ic con ol cos s.
P oblems in which he op imize s y o minimize o maximize he expec ed alue o
a ce ain cos unc ion un il a gi en andom e en occu s a e known as homing p oblems.
Whi le [
8
] conside ed he case when he op imize con ols an
n
-dimensional di usion
p ocess un il i lea es a gi en subse o
Rn
. Rishel [
9
] also ea ed he homing p oblems o
n
-dimensional di usion p ocesses; hese p ocesses we e mo e ealis ic models o he wea
o de ices han hose p oposed by a ious au ho s.
The au ho has ecen ly ex ended homing p oblems o he op imal con ol o queueing
sys ems in con inuous ime; see [
10
–
12
]. In hese h ee pape s, he aim was o de e mine
he op imal numbe o se e s wo king a ime
. He also published a pape ([
13
]) on a
homing p oblem o a con olled andom walk wi h wo op imize s.
Nex , we de ine he alue unc ion
F(k) = min
(un, n)
0≤n≤τ(k)−1
E[J(k)]. (5)
The unc ion
F(k)
is he expec ed cos incu ed (which can some imes be a ewa d) i
bo h op imize s choose he op imal alue o
un
and
n
be ween he ini ial ime
n=
0 and
ime τ(k)−1.
In Sec ion 2, he dynamic p og amming equa ion sa is ied by he alue unc ion
F(k)
will
be de i ed, and a pa icula p oblem will be sol ed explici ly. In Sec ion 3, he p oblem
o mula ion will be modi ied. We will hen assume ha he alue o
un
is known, and we
will look o he alue o
n
ha maximizes he expec ed alue o a ce ain cos unc ion.
Concluding ema ks will be made in Sec ion 4.
2. Dynamic P og ammic Equa ion
We will de i e he dynamic p og amming equa ion sa is ied by F(k). We ha e
Games 2024,15, 19 3 o 8
F(k) = min
(un, n)
0≤n≤τ(k)−1
q1u2
0+q2 2
0+λ( 0−u0)(6)
+Eτ(k)−1
∑
n=1nq1u2
n+q2 2
n+λ( n−un)o.
Then, making use o Bellman’s p inciple o op imali y (see [14]), we can w i e ha
F(k) = min
(u0, 0)q1u2
0+q2 2
0+λ( 0−u0) + E[F(X1(1))]. (7)
Indeed, wha e e he wo op imize s decide o do a ime
n=
0, he decisions hey
make om ime n=1 o ime τ(k)−1 mus be op imal.
Rema k 2. Equa ion (7) is alid because o ou assump ions ha
Ai
, o
i=
1, 2, and
S
ha e a
geome ic dis ibu ion. Indeed, as is well known, his dis ibu ion possesses he memo yless p ope y.
I we assume ins ead ha hese andom a iables ha e any o he (disc e e) dis ibu ion, hen we will
ha e o ake he pas in o accoun , ende ing he op imiza ion p oblem almos in ac able.
Fu he mo e, we ha e
E[F(X1(1))]=F(k+1)pA(0)(1−pS) + F(k−1)[1−pA(0)] pS(8)
+F(k)pA(0)pS+ [1−pA(0)](1−pS).
Hence, we can s a e he ollowing p oposi ion.
P oposi ion 1. The alue unc ion F(k)sa is ies he dynamic p og amming equa ion (DPE)
F(k) = min
(u0, 0)q1u2
0+q2 2
0+λ( 0−u0)(9)
+F(k+1)pA(0)(1−pS) + F(k−1)[1−pA(0)] pS
+F(k)pA(0)pS+ [1−pA(0)](1−pS).
Mo eo e , we ha e he bounda y condi ion F(k) = 0i k <j o k = .
Rema k 3.
(i)
We ake o g an ed ha each op imize does no know wha he o he has decided o do. In
Sec ion 3, we will assume ha Company 2 knows he decision made by Company 1.
(ii)
The e a e ou possibili ies o
(u0
,
0)
:
(
0, 0
)
,
(
0,
−pA)
,
(pA
, 0
)
and
(pA
,
−pA)
. I we sol e
he di e ence equa ion co esponding o each possible alue o
(u0
,
0)
, we ac ually ob ain
he alue o he unc ion
F(k)
i he op imize s choose he same alue o
(u0
,
0)
o any
k
.
Hence, we canno ob ain he alue unc ion and/o he op imal con ols o any alue o
k
by
compa ing he ou exp essions o F(k)ob ained by sol ing he ou di e ence equa ions.
(iii) We can w i e ha
(un
,
n) = (un(k)
,
n(k))
. The numbe o possible pai s
(u0(k)
,
0(k))
o
j≤k<
is equal o 4
−j
. I we ha e he alues o
(u0(k)
,
0(k))
o
k=j
,
. . .
,
−
1, we can
sol e a sys em o
−j
linea equa ions o ob ain he co esponding alues o
F(k)
o any
k
. I
−j
is small, i is a simple ma e o conside all he possible alues o
(u0(k)
,
0(k))
and
compu e he unc ion
F(k)
o
k=j
,
. . .
,
−
1. We can hen de e mine he op imal con ols
and he associa ed alue unc ion.
In he ollowing subsec ion, a pa icula p oblem will be sol ed explici ly.
Games 2024,15, 19 4 o 8
An Example
Suppose ha
q1=q2=λ=
1,
j=
1 and
=
3. Mo eo e , we ake
pA=
1
/
4 and
pS=
1
/
2. The alues o
F(
1
)
and
F(
2
)
o he 4
2=
16 possible choices o
(u0(
1
)
,
0(
1
))
and (u0(2), 0(2)) a e p esen ed in Table 1.
Table 1. Values o F(1)and F(2) o he 16 possible choices o he con ol a iables.
No. u0(1) 0(1)u0(2) 0(2)F(1)F(2)
1 0 0 0 0 0 0
2pA000−0.6 −0.45
3 0 −pA0 0 −0.375 −0.28125
4 0 0 pA0−0.107 −0.4286
5 0 0 0 −pA−0.125 −0.5
6pA−pA0 0 −0.923 −0.692
7pA0pA0−0.75 −0.75
8pA0 0 −pA−1.125 −1.5
9 0 −pApA0−0.375 −0.5625
10 0 −pA0−pA−0.375 −0.75
11 0 0 pA−pA−0.231 −0.923
12 pA−pApA0−0.964 −0.857
13 pA−pA0−pA−1.125 −1.5
14 pA0pA−pA−1.2 −1.65
15 0 −pApA−pA−0.375 −1.03125
16 pA−pApA−pA−1.154 −1.615
Fo example, i
(u0(
1
)
,
0(
1
)) = (
0, 0
)
and
(u0(
2
)
,
0(
2
)) = (
0,
−pA)
(Case no. 5),
hen we ha e
pA(
0
) = pA
i
k=
1 and
pA(
0
) =
0 i
k=
2. We mus sol e he sys em o
linea equa ions
F(1) = 1
8F(2) + 1
2F(1)(10)
F(2) = −3
16 +1
2F(2) + 1
2F(1), (11)
whose solu ion is F(1) = −1/8 and F(2) = −1/2.
We see ha he op imal s a egy is o choose Case no. 14; ha is, we ake
(u0(
1
)
,
0(
1
))
=(pA, 0)and (u0(2), 0(2)) =(pA,−pA).
Rema k 4. The ou di e ence equa ions ha mus be sol ed, subjec o he bounda y condi ions
F(0) = F(3) = 0, a e gi en below. We deno e hei solu ions by Fi(k), o i =1, 2, 3, 4.
(1) (u0, 0) = (0, 0):
F(k) = 1
8F(k+1) + 3
8F(k−1) + 1
2F(k). (12)
We easily ind ha he solu ion is F1(k)≡0.
(2) (u0, 0) = (pA, 0):
F(k) = −3
8+1
4F(k+1) + 1
4F(k−1) + 1
2F(k). (13)
We ind ha
F2(k) = 3
8k(−3+k). (14)
(3) (u0, 0) = (0, −pA):
F(k) = −3
8+1
2F(k−1) + 1
2F(k). (15)
Games 2024,15, 19 5 o 8
Because
pA(
0
) =
0,
X1(n)
canno each he alue 3 i
k=
1 o 2. The solu ion ha
sa is ies he bounda y condi ion F(0) = 0 is
F3(k) = −3
8k. (16)
(4) (u0, 0) = (pA,−pA):
F(k) = −3
8+1
8F(k+1) + 3
8F(k−1) + 1
2F(k). (17)
We ha e
F4(k) = 3
52 (−3+3k+1−26k). (18)
The unc ions
Fi(k)
, o
i=
2, 3, 4, a e shown in Figu e 1. Mo eo e , he alues o
Fi(k)
o k=1, 2 and i=1, 2, 3, 4 a e p esen ed in Table 2.
Figu e 1. Func ions F2(k)(solid line), F3(k)(do ed line) and F4(k)(dashed line) o k∈[1, 2].
Table 2. Values o Fi(k) o k=1, 2 and i=1, 2, 3, 4.
i Fi(1)Fi(2)
1 0 0
2−0.375 −0.75
3−0.75 −0.75
4−1.154 −1.615
No ice ha
F1(k)
co esponds o Case no. 1, while
F4(k)
co esponds o Case no. 16.
We obse e ha none o he unc ions
F1(k)
,
. . . F4(k)
is he alue unc ion. Howe e ,
F4(k)
yields alues which a e qui e close o hose ob ained wi h he alue unc ion. The e o e, i
−j
is la ge, so ha he numbe o equa ions o conside is also la ge, hen a subop imal
solu ion can be ob ained by assuming ha (u0(k), 0(k)) will be he same o any k.
3. Op imal Con ol When unIs Known
In his sec ion, we assume ha Company 2 knows he s a egy o Company 1, and
ies o maximize he expec ed alue o he ollowing cos unc ion:
C(k) =
τ(k)−1
∑
n=0
{q1un+q2 n}+K[X1(τ(k))]. (19)
The e minal cos unc ion K(·)is de ined by
K[X1(τ(k))] = K1i X1(τ(k)) = ,
K2i X1(τ(k)) = j−1, (20)
Games 2024,15, 19 6 o 8
whe e
K1<
0 and
K2>
0. The cons an
∈N
could be he maximum numbe o o de s
ha Company 1 can p ocess a he same ime.
Suppose ha Company 1 uses he con ol
un(k) = pA
o any
n
and any
k
. Company 2
mus decide whe he o choose n(k) = 0 o −pA. We de ine he alue unc ion
V(k) = max
n, 0≤n≤τ(k)−1E[C(k)]. (21)
P oceeding as in he p e ious sec ion, we can p o e he ollowing p oposi ion.
P oposi ion 2. The alue unc ion V(k)sa is ies he dynamic p og amming equa ion
V(k) = max
0∈{−pA,0}q1pA+q2 0+V(k+1)pA(0)(1−pS)
+V(k−1)[1−pA(0)] pS
+V(k)pA(0)pS+ [1−pA(0)](1−pS),
whe e
pA(
0
) =
2
pA+ 0
. Fu he mo e, he unc ion
V(k)
is such ha
V( ) = K1
and
V(j−
1
) = K2
.
The e a e now 2
−j
possible s a egies o Company 2. I
−j
is small, we can p oceed
as in he p e ious sec ion and calcula e he expec ed alue o
C(k)
o each possible s a egy.
Assume ha
q1=q2=
1,
pA=
1
/
4,
pS=
1
/
2,
K1=−
10,
K2=
10,
=
5 and
j=
1.
We p esen in Table 3 he alue o
V(
1
)
,
. . .
,
V(
4
)
o each o he 16 possible s a egies
ha Company 2 can choose. These alues a e ob ained by sol ing he sys em o ou
linea equa ions
V(k) = 1
4+ 0(k) + V(k+1)1
2+ 0(k)1
2(22)
+V(k−1)1
2− 0(k)1
2+V(k)1
2,
o
k=
1, 2, 3, 4, oge he wi h he bounda y condi ions
V(
0
) = −
10 and
V(
5
) =
10.
We conclude ha he op imal s a egy is he one ha co esponds o Case no. 15; ha is,
0(1) = 0 and 0(k) = −pA o k=2, 3, 4.
Table 3. Values o
V(
1
)
,
. . . V(
4
)
o he 16 possible choices o he con ol a iables
0(
1
)
,
. . .
,
0(
4
)
.
No. 0(1) 0(2) 0(3) 0(4)V(1)V(2)V(3)V(4)
1 0 0 0 0 8 5 1 −4
2−pA0 0 0 8.92 5.69 1.46 −3.77
3 0 −pA0 0 9.36 7.73 2.82 −3.09
4 0 0 −pA0 9.56 8.11 5.67 −1.67
5 0 0 0 −pA9.29 7.57 4.86 1.14
6−pA−pA0 0 9.45 7.81 2.81 −3.06
7−pA0−pA0 9.52 8.08 5.64 −1.68
8−pA0 0 −pA9.42 7.68 4.95 1.21
9 0 −pA−pA0 10.13 9.26 6.65 −1.17
10 0 −pA0−pA10 9 6 2
11 0 0 −pA−pA10.47 9.93 8.40 3.80
12 −pA−pA−pA0 9.72 8.87 6.31 −1.34
13 −pA−pA0−pA9.67 8.69 5.76 1.82
14 −pA0−pA−pA9.84 9.35 7.86 3.40
15 0 −pA−pA−pA10.49 9.98 8.44 3.83
16 −pA−pA−pA−pA9.83 9.34 7.85 3.39
Games 2024,15, 19 7 o 8
Rema k 5. As in he p e ious sec ion, we can ob ain a leas a subop imal solu ion which is close
he op imal one by sol ing he di e ence equa ions ob ained by assuming i s ha
0(k)≡
0,
namely
V(k) = q1pA+V(k+1)2pA(1−pS) + V(k−1)(1−2pA)pS(23)
+V(k)2pApS+ (1−2pA)(1−pS),
and hen ha 0(k)≡ −pA:
V(k) = q1pA−q2pA+V(k+1)pA(1−pS) + V(k−1)(1−pA)pS(24)
+V(k)pApS+ (1−pA)(1−pS).
The i s equa ion co esponds o Case no. 1, and he second one o Case no. 16. We ind ha
he solu ions o hese equa ions in he pa icula case conside ed abo e a e, espec i ely,
V1(k) = 10 −3
2k−1
2k2(25)
and
V16(k) = 1220
121 −10
121 3k. (26)
See Figu e 2.
Figu e 2. Func ions V1(k)(solid line) and V16(k)(do ed line) o k∈[1, 4].
No ice ha he solu ion ob ained when
0(k)≡
0(Case no. 1) ac ually gi es he minimum
o he expec ed alue o he cos unc ion
C(k)
. Mo eo e , we see in Table 3 ha he choice ha
co esponds o Case no. 11 almos yields he op imal solu ion ha we a e looking o .
4. Conclusions
In his pape , a homing p oblem o a queueing model in disc e e ime has been
conside ed. The p oblem can be seen as a dynamic game because he e a e wo op imize s
wi h opposing objec i es.
In Sec ion 2, dynamic p og amming was used o de i e he equa ion sa is ied by
he alue unc ion. F om his equa ion, one can deduce he op imal alues o he con ol
a iables. Howe e , we ha e seen ha , in o de o do his, one has o sol e a possibly la ge
numbe o sys ems o linea equa ions, subjec o he app op ia e bounda y condi ions.
Al hough sol ing each sys em is s aigh o wa d, epea ing his p ocedu e a la ge numbe
o imes can become edious. We ha e also seen ha i is possible o ob ain a good
subop imal solu ion o ou p oblem ai ly quickly.
In Sec ion 3, he p oblem o mula ion was modi ied. We assumed ha he s a egy o
Company 1 was known, and we looked o he s a egy ha Company 2 should adop o
Games 2024,15, 19 8 o 8
maximize he expec ed alue o a ce ain cos unc ion. We ea ed he case when he con ol
a iable
un
is always equal o
pA
. Howe e , he same ype o analysis could be ca ied
ou o any choice o
un
. In pa icula , we could ind he op imal s a egy o Company 2 i
un≡0.
In heo y, we could easily ex end he p oblems conside ed o he case when each
op imize can choose be ween mo e han wo possible alues o he a iable i con ols.
The calcula ions would, howe e , become qui e complex. One could possibly use nume ical
simula ions o de e mine he op imal solu ions. Indeed, ins ead o sol ing a la ge numbe o
di e ence equa ions, simula ing he p oposed model can enable us o de e mine he op imal
solu ion by compu ing he alue unc ion o each simula ion. Simula ing geome ic
andom a iables is no a di icul ask.
Finally, i is also possible o conside op imal con ol p oblems o queueing models in
con inuous ime wi h wo op imize s.
Funding: This esea ch was suppo ed by he Na u al Sciences and Enginee ing Resea ch Council o
Canada.
Da a A ailabili y S a emen : No da a was used o his esea ch.
Acknowledgmen s: The au ho wishes o hank he anonymous e iewe s o his pape o hei
cons uc i e commen s.
Con lic s o In e es : The au ho epo s ha he e a e no compe ing in e es s o decla e.
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Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
