an inventory system with retrial demands and working vacation
TRANSCRIPT
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An Inventory System with Retrial Demands and
Working Vacation
J. Kathiresan*, . An!a"hagan*and K. Jeganathan**
"De#artment of $athematics, %la&a##a 'ni(ersit), *arai+udi,India""Ramanu-an Institute for %d(anced Stud) in $athematics, 'ni(ersit) of $adras, .he#au+, .hennai, India
Abstract# /his article considers a continuous re(ie retrialin(entor) s)stem at a ser(ice facilit), herein an item demanded
b) a customer is issued after #erformin& ser(ice on the item /he
arri(al time #oints of customers form a Poisson #rocess /he
in(entor) re#lenished accordin& to an , Qs #olic) and thelead times are assumed to follo an e3#onential distribution /he
demands that occur durin& the stoc+ out #eriod or the ser(er bus)
re&ular or or+in& (acation are #ermitted to enter into the orbitof infinite sie hen the in(entor) le(el is ero or no demands
in the s)stem or both, ser(er &oes to a or+in& (acation hich ise3#onentiall) distributed If the ser(er is in or+in& (acation or
the in(entor) le(el is ero, the im#atience occurs in orbitin&customers, that follos an e3#onential distribution /he -oint
#robabilit) distribution of the number of demands in the orbit,
the in(entor) le(el and the ser(er status is obtained in the stead)
state case Some s)stem #erformance measures are deri(ed, thelon&run total e3#ected cost rate is calculated and the results are
illustrated numericall)
Index Terms .ontinuous re(ie in(entor) s)stem, , Qs
Polic), Positi(e leadtime, Retrial demand, or+in& (acation
A$S %lassi&ication' 60705, 80J29
I IN/R:D'./I:N
he conce#t of ser(er (acation in in(entor) ith to ser(ers
as first introduced b) Daniel and Ramanara)anan ;1
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rates are b
and v
F b , hen the s)stem is in re&ular
and or+in& (acation, res#ecti(el), hich are e3#onentiall)
distributed /he ser(er ta+es a or+in& (acation at times henno customers in the s)stem or the in(entor) le(el is ero or both
or+in& (acation durations are e3#onentiall) distributed ith
#arameter %t the com#letion time of the or+in& (acation,
the ser(er sitches a re&ular bus) ie ser(ice rate from v
to
b if there are customers #rimar) or retrial in the s)stem:therise, the ser(er continues the or+in& (acation /heorbitin& customers ma) either retr) or ma) lea(e the orbit /he
lea(in& orbitin& customers are described as im#atientrene&in&
customers If the ser(er is in or+in& (acation or the in(entor)
le(el is ero, the im#atience occurs in orbitin& customers %n
im#atient customer lea(es the orbit inde#endenl) after a random
time hich is distributed e3#onentiall) ith #arameter0E
e assume the constant retrial #olic) for these orbi
demands, that is #robabilit) of a re#eated attem#t of an orbitin&
demand is inde#endent of the number of demands in the orbit
Retrial re=uests from the orbit follo an e3#onential distribution
ith #arameter 0E %s the , Qs re#lenishment #olic)hen the onhand in(entor) le(el dro#s to a #refi3ed le(el, sa)
0Es an order for 1EG + ssSQ units is #laced /he#ositi(e lead time is e3#onentiall) distributed ith #aramete
0E e assume that the interdemand times beteen the
#rimar) demands, the lead times, retrial demand times, ser(er
re&ular #eriods and ser(er or+in& (acation #eriods are mutuall)
inde#endent random (ariables
otations
ijArom the assum#tions made on the in#ut and out#ut #rocesses, it can be shon that the tri#let{ }0,,, ttYtLtX
ith the state s#ace E is a $ar+o( #rocess
/o determine the infinitesimal &enerator
,,,,,G nljmkipP,
Enljmki ,,,,,
of this #rocess e use the folloin& ar&uments
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Cet 0Gi , Sk ,1,2,G
L an) arri(in& #rimar) demand ta+es the state of the #rocess from ,0, ki to ,1, ki ith the intensit)
L an) arri(in& #rimar) demand ta+es the state of the #rocess from,, mki
to ,1, mki + ith theintensit) , 1,!Gm
L /he ser(er chan&es from or+in& (acation to re&ular ta+es the state of the #rocess from,1, ki
to
,!, ki ith the intensit)
L /he com#letion of ser(ice from #rimar) demand ma+es a transition from,1, ki
to1,0, ki ith the
intensit) v
L /he com#letion of ser(ice from #rimar) demand ma+es a transition from ,!, ki to 1,0, ki ith the
intensit) b
Cet 0i , 0Gk
L an) arri(in& #rimar) demand ta+es the state of the #rocess from ,0, ki to ,01, ki + ith the intensit)
Cet 1i , Sk ,1,2,G
L an) arri(in& #rimar) demand ta+es the state of the #rocess from ,, mki to 1,, +mki ith the intensit) , mG0, 2
L an) arri(in& #rimar) demand ta+es the state of the #rocess from,, mki
to ,1, mki + ith theintensit) , mG1, !
L /he ser(er chan&es from or+in& (acation to re&ular ta+es the state of the #rocess from ,, mki to2,, +mki ith the intensit) , mG0, 1
L a retrial re=uests ta+es the state of the #rocess from,, mki
to1,1, + mki ith the intensit) ,
mG0, 2
L an im#atient customer ta+es the state of the #rocess from,, mki
to,1, mki
ith the intensit)
,
mG0, 1
L an im#atient customer ta+es the state of the #rocess from,0,0i
to1,0,0 i
ith the intensit)
L /he com#letion of ser(ice in the s)stem ma+es a transition from ,1, ki to 1,0, ki ith the
intensit) v
Cet 1i , Sk ,2,!,G
L /he com#letion of ser(ice in the s)stem ma+es a transition from,!, ki
to1,2, ki
ith the
intensit) b
L /he com#letion of ser(ice in the s)stem ma+es a transition from ,1,!i to ,0,0i ith the intensit)
b
Cet 0i , sk 1,2,G
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L a transition from ,, mki to ,, mQki + for 0,1,2,!Gm ta+es #lace ith the intensit) hen are#lenishment occurs
L a transition from ,0,0i to ,0, Qi ta+es #lace ith the intensit) hen a re#lenishment occurs
e obser(e that no transition other than the abo(e is #ossible
>inall), the (alue of ,,,,, mkimkip is obtained b)
11,,21,,,22G11,,21,,,22
1,,21,,2
nljmkipmkimkipnlj
nljmki
Mence e ha(e,,,,,, nljmkip
G
+
+
+
+
0,2G,,1,2,G1,
1G,G,G
0G0,G0,
G,G1,G
1,!G,,1,2,G0,G,G1,G
0G,,1,2,G0,G
1G,G,G,
mSki
mnklij
or
mki
mnklij
or
mSkimnklij
or
mSki
mnklij
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==
==
=
+
+
+
00
,
,1
,1
0,1G
,,1,2,G1,
G,G1,G,
0,2G,,1,2,G1,1G,G1,G,
0,1G,,1,2,G1,
2G,G,G
1G,,1,2,G0,G
2G,G,G,
!G1,G1,
0G1,G,G
!G,,2,G1,
1G1,G,G
!G,,1,2,G0,G
0G1,G,G,
1G,,1,2,G0,
0G1,G,G,
m
mn
k
kl
i
ij
or
m
Ski
mnklij
mSki
mnklij
mSki
mnklij
or
mSki
mnklij
mki
nklij
or
mSki
mnklij
or
mSki
nklij
mSki
nklij
b
v
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=++=+++
++
+++
+
++
+
++++
++
+
+
0
G
,,,2,1
,G
,1
,G,
2G,,1,2,G0,G
G,G,G,
!G,,1,2,G0,
G,G,G,
1G,,1,2,G0,G
G,G,G,
0G,,0,1,2,G0,G
G,G,G,
!G,,2,1,G0,
G,G,G,
1G,,2,1,G0,G
G,G,G,
0G,,2,1,G0,G
G,G,G,
0G0,G0,
G,G,G
0,1,2,!G,,1,2,G0,
G,G,G,
m
mn
Sssk
kl
i
ij
mski
mnklij
mski
mnklij
mski
mnklij
mski
mnklij
mSsski
mnklij
mSsski
mnklij
mSsski
mnklij
mki
mnQklij
or
mski
mnQklij
b
v
b
v
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++
++
++++
++++
+++
+++++
otherise0,
0G0,G1,G,G,G,
2G,,1,2,G1,
G,G,G,
1G,,1,2,G1,
G,G,G,
0G,,1,2,G1,
G,G,G,
2G,,2,1,G1,
G,G,G,
1G,,2,1,G1,
G,G,G,
mkimnklij
mski
mnklij
mski
mnklij
mski
mnklij
mSsski
mnklij
mSsski
mnklij
v
v
Denotin&,!,,2,,,1,,,0,,,,1,!,,1,2,,1,1,,,1,0,0,0,G SqSqSqSqqqqqqq fo
0,1,Gq 7) orderin& states le3ico&ra#hicall), the infinitesimal &enerator P can be con(enientl) e3#ressed in a bloc+#artitioned matri3 ith entries
+
otherise0,
2,1,G1,G,
2,1,0,G1,G,
2,1,G,G,
0G,G,
G
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LAM
LAM
LAM
LA
P00
00
00
000
G
1
here
+
+
==
+
++
otherise0,
0G,G,
,,21,G,G,
11
,,2G1,G,
0G,G,
,,2,1G,G,
,2,1,G,G,
G
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klJ
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ith
klF
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kl!
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ijQ
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kl"
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.learl) 4!10 ,,, """"
and C are s=uare matrices of order 4 /he matri3 4" is of sie 4 1 and the matri3 1C isof sie 1 4 /he matri3 2" is of sie 1 1
Cet + be the stead)state #robabilit) (ector of Q /hat is, + satisfies
1G0,G +e+Q
/he (ector + can be re#resented b)
+ ,,, 10 S
+++
here0
+ G 0,0
i+ G ,,,
,!,2,1,0 iiii , Si ,1,2,G
-heorem #he ste%&'(st%te prob%bilit' ve)tor + )orespon&in* to the *ener%tor Q is *iven b'
1,G0,G21
4
i"" ii
+
1,,!,2,G0,G11
!
++ si"" ii
,,,!2,G0,G01
!
Qssi""
ii+++
1G0,G11
0
1
!
+++ QiC"" Qiii
,,,!2,G0,G101
!
SQQiC"" Qiii ++++
0G
0
C"
sS +
roo&'e ha(e0GQ+ , 1G+e
%fter lon& sim#lications, the abo(e e=uations, e3ce#t the last one, )ields
++
+
+
+
+
++
+
SQQi"
C
"
"
"
"
"
"
"
"
"
"
"
"
"
"
Qi"
C
"
"
"
"
"
"
Qssi"
"
"
"
"
"
si"
"
"
"
QikQikQi
k
Qisis
i
sis
i
sis
i
i
i
i
,!,2,G,
1
1G,1
,!,2,G,1
1,2,1,G,1
G
!
0
1
!
0
1
!
1
4
21
1G
1
!
0
1
!
0
!
1
4
20
!
1
1
!
0
!
1
4
20
1
!
0
!
1
4
20
1
!
1
4
20
here0 can be obtained b) sol(in&,
0G
0
C"
sS +and
1G2
0G
ei
S
i
,
that is
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0G1
1
1
!
1
4
2
0
!
1
!
0
1
!
1
4
21
1G
1
!
0
1
!
0
!
1
4
2
+
+
+
s
s
QSkQSkQS
k
QS
S
sSs
C
"
"
"
"
""
C
"
"
"
"
"
"
"
"
"
"
"
"
"
"/(0+
and
+
+
+
+
+
1
!
0
!
1
4
2
2G
1
!
1
4
21
1G
11
sis
iQ
si
i
is
i "
"
"
"
"
"
"
"
"
"I
/(0+
+
+
+
+
+ 1
!
0
!
1
4
2
2G!
1
!
0
!
1
4
21 11
sis
iS
Qi
sQs
Q
"
"
"
"
"
"
"
C
"
"
"
"
"
"
1G!
1
!
0
1
!
1
4
21
1G
1
!
0 e"
C
"
"
"
"
"
"
"
"QikQik
Qi
k
Qi
+
Ne3t, e deri(e the condition under hich the s)stem is stable
1emma #he st%bilit' )on&ition of the s'stem +n&er st+&' is *iven b'
ee E 11
0FFF
1
roo&'>rom the ell +non result of Neuts9 on the #ositi(e recurrence of P e ha(e
e+e+ LM E
and b) the e3#loitin& the structure of matrices M and L and + the stated result follos
).2 Steady state analysis
It can be seen from the structure of the rate matri3 P and from the lemma 1 that the mar+o( #rocess
1121,21,2I2 tYtLtX,
0Jt ith the state s#ace E is re&ular Mence the limitin& #robabilit) distribution
0J0,0,NG,G,GIlimG,,
YLXmtYktLitXprtmki
, e3ists and is inde#endent of the initial state
/hat is, G ,, 10
satisfies
0GP3 , 1G3e
e #artition the (ectori , for 0,1,2,Gi as follos
i G ,,, ,,1,0 Siii
hich is #artitioned as follos, for Sk1,0i G
,0,0i
, ki G ,,, ,!,,2,,1,,0, kikikiki
-heorem 2 ,hen the st%bilit' )on&ition -./ hol&s *oo&0 the ste%&' st%te prob%bilit' ve)tor 3 is *iven b'
0,1,2,G,G i1 ii /(0
3 2
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here the matri3 1 satisfies the matri3 =uadratic e=uation
0G2
L1AM1 ++ !
and the (ector/(0
3 is obtained b) sol(in&
0G 1 1MA+/(03sub-ect to normaliin& condition
1G 1e1I /(03
roo&'/he theorem follos from the ell +non result on matri3&eometric methods Neuts9
).) %om45tation o& R matri6
In this subsection e #resent an efficient al&orithm for com#utin& the rate matri3 1 hich is the main in&redient for discussin&
=ualitati(e beha(ior of the model under stud) /he 1 matri3 is of sie 4S O 1 can be com#uted b) usin& lo&arithmic reductional&orithm
1ogarithmic red5ction algorithm
Co&arithmic reduction al&orithm is de(elo#ed b) Catouche and Ramasami ;11< hich has e3tremel) fast =uadratic con(er&ence
Mere e discuss onl) the im#ortant ste#s in(ol(ed in this al&orithm e refer the reader to Catouche and Ramasami ;11< for more
details about this al&orithm
Ste4 (' LA" 1
, MAF
1 , F!G , and "#G
Ste4 '
F""F2 +G2
G"E
E2I" 1
2FE
E2IF 1 #L!! +
#"#
.ontinue Ste4 'until FNNNN !ee
Ste4 2'1G + L!AL1
IV SKS/B$PBR>:R$%N.B$B%S'RBS
In this section some #erformance measures of the s)stem under consideration in the stead) state are deri(ed7. 864ected inventory level
Cet i denote the a(era&e in(entor) #osition in the stead) state /hen[ ],,
!
0G1G0G
G kji
k
S
ji
i j
47.2 864ected reorder rate
Cet r
denote the e3#ected reorder rate in the stead) state /hen
[ ]1,!,1,1,0G
G ++
+ sibsi
vi
r 5
7.) 864ected n5m!er o& demands in the or!it
Cet o
denote the a(era&e in(entor) #osition in the stead) state /hen
[ ] [ ],0,01G
,,!
0G1G1G
G i
i
kji
k
S
ji
o ii
+8
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7.7 8&&ective reneging rate &or an or!iting c5stomer
Cet ro
denote the effecti(e rene&in& rate for an orbitin& customer in the stead) state /hen
[ ],,,0,01
0G1G1G
G kjii
k
S
ji
ro +
97.9 :verall rate o& retrials
Cet or
denote o(erall rate of retrials in the stead) state /hen
[ ] [ ],0,01G
,,!
0G1G1G
G i
i
kji
k
S
jior
+@
7.; -he s5ccess&5l retrial rate
Cet sr
denote successful retrial rate in the stead) state /hen
[ ],2,,0,1G1G
G jijiS
ji
sr +
67.< -he &raction o& s5ccess&5l rate o& retrial
Cet sr
denote successful retrial rate in the stead) state /hen
or
srfr
G
10
V /:/%CBPB./BD.:S/R%/B
/o com#ute the total e3#ected cost #er unit time, e consider the folloin& costs
s) Setu# cost #er order
h) /he in(entor) carr)in& cost #er unit item #er unit time
w) aitin& cost of a customer in the orbit #er unit time
r) rene&in& cost #er customer #er unit time
/he lon& run total e3#ected cost rate is &i(en b)
rorowrhis
))))Ss#C +++G,>rom e=uations 4,5,8 and 9, e obtain
[ ] +
++
++ 1,!,1,1,0G
,,!
0G1G0G
G, sibsi
v
i
h
kji
k
S
ji
s )j)Ss#C
[ ] ,,,0,01
0G1G1G
,0,0
1G
,,!
0G1G1G
++
+
kjii
k
S
ji
r
i
i
kji
k
S
ji
w )ii)
VI N'$BRI.%CICC'S/R%/I:NS
In this section, e discuss some numerical e3am#les indicates the function, Ss#C
to be con(e3>i&ure 1refers the chan&es
of s and S are ho to affect the total e3#ected cost rate /he table 1 #resents the total e3#ected cost rate for (arious combinations ofs and Sb) fi3in& the other #arameters and costs as 4,0G1@,G,5G 02G16,G199,G2,G bv and
1G04@,G@,!G006,G rwhs )))) /he sim#le numarical search #rocedure are used to obtain the o#timal (alues o
, Ss#Csa) ,
""" Ss#C /he o#timal (alue of the total e3#ected cost rate is20,55#C
G641942 /he o#timal cost foreach S is shon in underline and the o#timal cost for each s is bold Some of the results are #resented in /ables 2 throu&h 1! here
the loer entr) in each cell &i(es the o#timal total e3#ected cost rate and the u##er entries the corres#ondin&"S and
"s
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-a!le
'
-otal
e64ected cost rate as a &5nction o& S and s
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s
S
19 1@ 16 20 21 22 2!
52 64!448 642984 642990 64!400 64481@ 64841! 64@9605! =.7))>9 64249! 64221! 642540 64!419 644@26 648999
54 64!594 =.7279( 641648 641668 642582 64!828 6451@5
55 64!660 64289! =.7=79 =.7
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ISSN 2250!15!
4,0G1@,G,5G 02G16,G199,G2,G bv 1G04@,G@,!G006,G rwhs ))))
?ig. '% three dimensional #lot of the cost function, Ss#C
86am4le In this e3am#le e stud) the im#act of arri(al rate , ser(ice rates v
and b
, the lead time #arameter , the
retrial rate , and the rena&in& rate , on the o#timal (alues , ""Ss and the corres#ondin& total e3#ected cost rate
"#C 7)
fi3in& #arameter
02Gand the costs (alues as
1G048,G@,!G006,G rwhs ))))
e obser(e the folloin& from table 2to 9
1 /he total e3#ected cost rate increases hen increases and the total e3#ected cost rate decreases hen ,,
bv ,, increase
2 If , v
increase, then"S monotonicall) increases If ,, b
,increase, then
"S monotonicall) decrease
! If increases, then"s monotonicall) increases If ,, bv
,,increase, then
"s monotonicall) decreases
-a!le 2' 8&&ect o& arrival rate and sevice rate v
on the o4timal val5es
4,0G1@,G 16G2,G b
v
195 198 199 19@ 196
466 55 20 55 20 55 20 58 20 58 20
@6@@82 @69811 @68!91 @651!9 66!@6@
500 55 20 55 20 55 20 55 20 58 20
62002! 61@944 619495 618219 614656
501 58 21 58 21 58 21 58 21 58 21
64!021 641926 64044@ 6!6198 6!9615
502 58 21 58 21 58 21 58 21 58 21
68@191 688@50 6855!6 6842!6 68264@
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ISSN 2250!15!
50! 59 22 59 22 58 21 58 21 58 21
665956 6644!0 66!060 661956 6604!6
-a!le )' 8&&ect o& arrival rate and sevice rate b
on the o4timal val5es
4,0G1@,G 919G2,G v
b
1@@ 1@6 160 161 162
466 58 21 55 20 55 20 58 20 55 16
68!!94 629!95 @68!91 @86495 @45612
500 58 21 58 21 55 20 55 20 58 20
661251 651501 619495 @@@091 @82489
501 59 22 58 21 58 21 55 20 58 20
102166@ 699624 640449 60@2!2 @@0211
502 59 22 59 22 58 21 58 21 58 20
1058016 100900! 6855!6 6!0142 @66502
50! 59 22 59 22 58 21 58 21 58 21
1081!6@ 10!6092 66!060 654012 620521
-a!le 7' 8&&ect o& arrival rate and retrial rate on the o4timal val5es4,0G1@,G 16G9,19G vv
166 200 201 202 20!
466 55 20 55 20 55 20 55 16 55 16
61@425 @68!91 @98!08 @59648 @4100@
500 58 21 55 20 55 20 55 20 55 16
641545 619495 @6582! @95926 @5952!
501 58 21 58 21 55 20 55 20 55 20688@04 640449 618812 @6465! @95221
502 58 21 58 21 58 21 55 20 55 20
664546 6855!6 6!644! 615@!2 @64!58
50! 59 22 58 21 58 21 58 21 55 20
10250@5 66!060 684!96 6!@5!1 6151!5
-a!le 9' 8&&ect o& lead time and sevice rate v
on the o4timal val5es
4,0G5,G 16G2,G b
v
195 198 199 19@ 196
196 58 21 58 21 55 20 55 20 58 20
6200@8 61@@20 619558 618268 6150!!
1@0 55 20 55 20 55 20 55 20 58 20
62002! 61@944 619495 618219 614656
1@1 55 20 55 20 55 20 55 20 58 20
616644 61@888 619400 618144 614@@6
1@2 55 20 55 20 55 20 55 20 58 20
616@86 61@564 619!!0 618098 614@24
1@! 55 20 55 20 55 20 55 20 58 20
616@00 61@529 619284 618012 614984
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-a!le ;' 8&&ect o& lead time and sevice rate b
on the o4timal val5es
4,0G5,G 919G2,G v
b
1@@ 1@6 160 161 162
196 58 21 58 21 55 20 55 20 58 16
661!!5 651586 619558 @@@1!9 @82516
1@0 58 21 58 21 55 20 55 20 58 20
661251 651501 619495 @@@091 @82489
1@1 58 21 58 21 55 20 55 20 55 16
661192 6514!9 619400 @@@006 @82409
1@2 58 21 55 20 55 20 55 20 55 16
66106@ 651!91 619!!0 @@965! @82!!@
1@! 58 21 55 20 55 20 55 20 55 16
66102@ 651261 619284 @@9601 @82294
-a!le
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International Journal of Scientific and Research Publications, Volume 4, Issue 12, December 2014 22
ISSN 2250!15!
60!909 60946@ 611200 614602 61@524
00@@ 55 20 55 20 58 20 58 20 59 20
60595@ 606588 61!!01 61900! 620895
00@6 55 20 55 20 58 20 58 20 58 20
609@06 611819 615402 616104 622@08
0060 55 20 55 20 55 20 58 20 58 20
606@80 61!88@ 619695 621205 624609
0061 54 20 55 20 55 20 58 20 58 20
611@9! 615916 616528 62!!08 62900@
-a!le =' Sensitivity o& s)
and w)
on the o4timal val5es
1G@,!G rh ))
w)
s)
04! 044 045 048 049
00@9 58 20 58 20 58 20 58 20 58 20
@94@28 @@8650 @66095 611200 62!!25
00@@ 58 20 58 20 58 20 58 20 58 20
@98629 @@6051 601198 61!!01 625428
00@6 55 20 58 20 58 20 58 20 58 20
@960244 @6115! 60!299 615402 6295290060 55 20 55 20 55 20 55 20 55 20
@@1095 @6!206 605!42 619495 626806
0061 55 20 55 20 55 20 55 20 55 20
@@!128 @65280 609!6! 616528 6!1880
-a!le (' Sensitivity o& s)
and r)
on the o4timal val5es
048G@,!G wh ))
r)
s)
066 100 101 102 10!
00@9 58 20 58 20 58 20 58 20 58 20610600 611200 611500 611@00 612100
00@@ 58 20 55 20 58 20 58 20 58 20
61!001 61!!01 61!801 61!601 614201
00@6 58 20 58 20 58 20 58 20 55 20
615102 615402 615902 618002 618!02
0060 55 20 55 20 55 20 55 20 55 20
619195 619495 619998 61@098 61@!99
0061 55 20 55 20 55 20 55 20 55 20
616228 616528 616@29 62012@ 6,2042@
-a!le ' Sensitivity o& h)
and w)
on the o4timal val5es
1G06,0G rs ))
w)
h)
04! 044 045 048 049
!8 54 20 54 20 55 20 55 20 55 20
@9!445 @@55@@ @69929 606@80 62166!
!9 55 20 55 20 55 20 55 20 55 20
@9928@ @@6401 6015!4 61!88@ 625@01
!@ 55 20 55 20 55 20 55 20 55 20
@@1095 @6!206 605!42 619495 626806
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International Journal of Scientific and Research Publications, Volume 4, Issue 12, December 2014 2!
ISSN 2250!15!
!6 58 20 58 20 58 20 58 20 58 20
@@4@!1 @68655 6060@0 621205 6!!!!0
40 58 20 58 20 58 20 58 20 58 20
@@@5!! 600859 6129@2 624609 6!90!2
-a!le 2' Sensitivity o& h)
and r)
on the o4timal val5es
480G06,0G ws ))
r)
h)
066 100 101 102 10!
!8 55 20 55 20 55 20 55 20 55 20
606556 606@80 610181 610481 610982
!9 55 20 55 20 55 20 55 20 55 20
61!!89 61!88@ 61!68@ 614286 614586
!@ 55 20 55 20 55 20 55 20 55 20
619195 619495 619998 61@098 61@!99
!6 58 20 58 20 58 20 58 20 58 20
620605 621205 621505 621@05 622105
40 58 20 58 20 58 20 58 20 58 20
624809 624609 625221 625509 625@09
-a!le )' Sensitivity o& w)
and r)
on the o4timal val5es
@!G006,G hs ))
r)
w)
066 100 101 102 10!
04! 55 20 55 20 55 20 55 20 55 20
@@0995 @@1095 @@1!98 @@1899 @@1699
044 55 20 55 20 55 20 55 20 55 20
@6260@ @6!206 @6!506 @6!@10 @64111045 55 20 55 20 55 20 55 20 55 20
605041 605!42 60584! 60564! 608244
048 55 20 55 20 55 20 55 20 55 20
619195 619495 619998 61@098 61@!99
049 55 20 55 20 55 20 55 20 55 20
626!0@ 626806 626606 6!0210 6!0510
VII .:N.C'SI:N
e anal)sed an , Qs in(entor) s)stem ith retrial
customers and or+in& (acation Primar) inter arri(al times,retrial times, ser(ice times and or+in& (acation times areinde#endent e3#onentiall) distributed random (ariables e ha(e
deri(ed the stead) state distribution of the s)stem usin& $atri3
anal)tic methods and se(eral #erformance measures ha(e also
been calculated Some numerical solutions are #resented toillustrate the =ualitati(e beha(ior of the s)stem
%.*N:CBDQ$BN/
N %nabha&an?s Research as su##orted b) the Nationa
7oard for Mi&her $athematics D%B, Qo(ernment of India
throu&h research #ro-ect 2A4@11A2011ARD IIA1141
RB>BRBN.BS
;1< Daniel, J * and Ramanara)anan, R %n in(entor) s)stem ith to ser(erand rest #eriods .ahiers du .BR:, 'ni(ersite Cibre De 7ru3elles16@9, 26, 65100
;2< Daniel, J * and Ramanara)anan, R%n in(entor) s)stem ith rest #eriodsto the ser(er Na(al Research Co&istics, John ile) and Sons 16@@, !511612!
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