Applied Mathematics, 2011, 2, 403-409
doi:10.4236/am.2011.24049 Published Online April 2011 (http://www.SciRP.org/journal/am)
Stationary Characteristics of the Single-Server
Queue System with Losses and Immediate
Service Quality Control
Aleksey I. Peschansky
Sevastopol National Tech nical University, Sevastopol, Ukraine
Received December 27, 2010; revised January 31, 2011; accepted February 1, 2011
Abstract
Semi-Markovian model of operation of a single-server queue system with losses and immediate service qual-
ity control has been built. In case of unsatisfactory request service quality, its re-servicing is carried out.
Re-servicing is executed till it is regarded satisfactory. Time between request income, and request service
time are assumed to be random values with distribution functions of general kind. An explicit form of the
system stationary characteristics has been defined.
Keywords: Queue System, Semi-Markovian Process, System Stationary Characteristics, Request Service
Quality
1. Introduction
A large number of works [1-4] are dedicated to the queue
systems with losses. In most of them the incoming flux
of requests is supposed to be Poisson one or request ser-
vice time is considered to have exponential distribution.
This admission allows efficient modeling of the system
operation by means of Markovian processes. But if the
incoming flux of requests makes a renewal process and
request service time distribution is of general kind, es-
sential difficulties arise when defining system stationary
characteristics in explicit form. To overcome them an
apparatus of semi-Markovian processes with a common
phase field of states is used in the works [5,6]. Thus, in
[6] stationary characteristics of the single-server queue
system 10GI G with losses were found. In the pre-
sent article analogical characteristics for the same system
have been defined under the assumption that the server
admits both satisfactory and unsatisfactory request ser-
vice. In case of the latter one the re-servicing begins im-
mediately. It is repeated until the service is regarded sa-
tisfactory.
In the second chapter of the article the system opera-
tion is described, mathematical problem definition and
research purpose are stated. In the third section semi-
Markovian model of system operation is built, and sta-
tionary distribution of embedded Markovian chain is
given. In the fourth section system stationary characteris-
tics are defined. These are: final probabilities that the
server is free, is in state of service or re-servicing, and
mean dwelling times in these states. Besides, formulas of
the system stationary characteristics for some subcases
are given here. One of them is the exponential distribu-
tion of time between request income and request service
time.
2. The Problem Definition
Let us investigate the queue system (QS) 10GIG with
losses and a single server. Request service time is a ran-
dom value (RV)

with an absolutely continuous dis-
tribution function (DF)
 
F
tP t   and density


f
t. Time period between requests’ income is a RV

with an absolutely continuous DF



Gt Pt 


and density


g
t. If the server is busy with request ser-
vice, all the incoming requests are lost. With the proba-
bility p request service is regarded to be successful,
and the queue system passes into a standby state that
lasts till next request comes. With the probability
1qp

 the request service is considered unsatisfacto-
ry, and the server begins re-servicing immediately. Time
period of such a kind of service is a RV

with an ab-
solutely continuous DF
 
tP t 

 and density


t  . After re-servicing with the probability p the

A. I. PESCHANSKY 404
service is regarded satisfactory and with the probability
q the request is sent to re-servicing procedure . This pro-
cess lasts until the service is considered sufficient. It is
assumed that RV ,


and

are independent, have
finite mathematical expectations ,,
M
MM


and
variances ,,DDD


respectively. It is necessary to
define the following stationary characteristics of the sys-
tem: the final probabilities that the queue system is in a
standby state; that the system is busy with the primary
service or re-servicing; mean time periods of system’s
dwelling in the above-mentioned states.
3. Semi-Markovian Model Building
In order to build the model of the system operation an
apparatus of semi-Markovian processes with a discrete-
continuous phase field of states [6,7] is used. Let us de-
scribe the system operation with the help of semi- Marko-
vian process (SMP)

t

with a phase field

21,10,21,22,12Exxxx.
Let us write out the codes of the states:
21 – service of the request that has come begins;
10x – request service has been successfully co mpleted,
the server passes into a standby state that lasts time x (till
the next request income);
21x – the incoming request has been lost, the server is
busy with the primary service that will last time x;
22x – the incoming request has been lost, the server is
busy with re-servicing that will last time x;
12x – request service has been completed, and its re-
servicing has begun, time x is left till the next request
income.
In Figure 1 time diagram of the system operation is
shown, and in Figure 2 there is the system transition
graph.
System dwelling times in the states are defined by the
formulas:
211021 2212
,, ,
xxx x
x
xx

 

,
where

is a sign of minimum.
Let us define the probabilities and probability densities
of the embedded Markovian chain (EMC)


,0
nn
 
transitions:
 
 
21 10
21 21
00
12 21
21 10
0
d, d,
d, 1,
xx
xx
pgtfxtt ppftgxtt
pqftgxttP


 






21 10
21 21
12
21
,; ;
;
xx
yy
x
y
pgyxxyppgyx
pqgyx
 




22 10
22 22
12
22
,; ;
;
xx
yy
x
y
pgyxxyppgyx
pqgyx
 




10 22
12 12
12
12
,; ;
,.
xx
yy
x
y
ppyxxypyx
pqyxxy


 

Now we can proceed to EMC stationary distribution
definition, the system of integral equations for its defini-
tion is the following one:
 
  
  
  

21
0
21 0
21 0
21 21
0
2121dd ,
2222d12d ,
1212dd2122d ,
1012dd2122d ,
10d,10
xx
x
xx
xx
xgyx yygyxfyy
x gyxyyyxyy
x
qyxy yqfyxgy yqgyxyyy
x
py xyypfy xgyypgyxyyy
xx


 
 
 





 
 
  








0
122122d 1.xxxxx 















 



(1)
Let us introduce the following integral operators:
   
d, d,d,
g f
xxx
A
gyxyyAyx yyAfyx yy



 

 
000
d, d,d.
g f
A
gyx yyAyx yyAfyx yy



 


A. I. PESCHANSKY 405
Figure 1. Time diagram of the system operation.
Figure 2. System transition graph.
Then the system of Equations (1) can be rewritten in
such a way—Equatio n ( 2).
We shall exclude

21
x
 and

22
x
 from the first
and second equations of the system (2) respectively, and
then substitute it in the third equation. The result is
  
  
1
21
1
21 ,
2212 ,
gg
g
x
IA Afx
xIAA x 






  


1
1
21
12 12
.
gg
gf
fg
xqAAIA Ax
qAAIAAgx














(3)
Here

 
1d,
gg
x
I
Axxhyxyy 



 [5]



1
n
gn
hxgx


 is the density of renewal function


g
H
x gene rat ed by DF

Gx.
Let us indicate
K

and
f
K
integral operators


1
,
0
,d,
gg
q
KqAAIAA
kxy yy


















1
,
0
,d.
gf
ff g
qf
KqAAIAA
kxy yy













With regard to the introduced operators the Equation
(3) will have the form




21
12 12.
f
x
KxKgx

  (4)
Let us write down kernels of integral operators
K

,
f
K
in explicit forms and single out their probability
sense

 

0
,
0
,d, ,
,
,d, ,
g
q
g
qyxq tyvtxtyx
kxy
qtyvtxtyx







 









 

0
,
0
,d, ,
,
,d, .
g
qf
g
qfyx qftyvtxtyx
kxy
qftyvtx tyx




 








 


 
 
 


21
21
21
21 0
21 0
21 21,
222212 ,
12212122,
10122122,
10d,
10122122d 1.
gg
g
g
f
g
f
xA xAfx
xAxA x
xqA xqAgxqAxx
x
pA xpAgxpAxx
xx
xxxxx





 
 

   





 






 









 






(2)

A. I. PESCHANSKY 406
Here
 
0
,d
t
gg
v
txgt xgt xuhu u 
 is the
density of the direct residual time t

for the renewal
process generated by RV

1
:ttt





 [8]. The func-
tions


,, q
kxy

and

,
0
,d
qf
kxygyy

 are densities
of probabilities of system transition from the state 12y to
the nearest state 12x and from the state 21 to the nearest
one 12x respectively.
As
 
 

,
0000
,ddd,d
1,
y
qg
kxyxq yxxqxtyvtxt
qy yq



 

it is not difficult to ensure that the operator
K

is the
operator of contraction in the space of summable func-
tions. That is why the solution of Equation (4) can be
found by means of successive approximations. This solu-
tion with regard to the identity

 
1
0
,d ggg qAIAfxqvyxfyy




can be represented in the form
 
21 1
12 ,
nn
n
x
ql x 


 (5)
where
 
10
,d
g
lxv yxfyy

 is the density of RV
that is time period between the end of primary request
service and the moment of the next request income. The
function



1
0
,d,2
n
l
ng
lxvyx yyn 



 is the den-
sity of random time period between the end of the

1
s
t
n request re-servicing and the moment of the
next request income. Here

 


11
10
,d
nn
y
ll
gn g
vyxlyxhyuguxu




is the density of the direct residual time for the renewal
process


1n
l
g
H
x
 generated by the functions
 
11
0
d
x
nn
Lx ltt

 and


Gx. Let us note that the
density of renewal function complies with the ratio

 
111
0
d.
n
x
l
gnn g
hxlxlxuhuu



The rest of EMC stationary distributions are defined
by the formulas
 
21 1
10 ,
nn
n
p
x
ql x
q




 
21
21d ,
g
x
x
hyxfyy 



 

21 10
21 100
22 d
dd.
nn
n
nn
n
xqxylyy
qy xyslss












(6)
The stationary probability 21

of EMC’s dwelling in
the state 21 is found with the help of normalization re-
quirement:
 


1
21 1
00
2d d.
n
l
n
gg
n
fxHxxqsHs s 






 






4. Definition of System Stationary
Characteristics
Mean values of system dwelling times in the states are
defined by the formulas
 
 
21 10
0
21 2212
00
d, ,
d, d.
x
xx
xx x
MFtGttMx
M
MGttM tt

 


 


(7)
Let us consider the following disjoint subsets of sys-
tem states:


010Ex,

121,21Ex,


212, 22Exx. Dwelling in the state 0
E means that
the server is free and the system is in a standby mode.
Dwelling in the state 1
E or 2 E signifies that the server
is busy with the primary request service or with re-ser-
vicing respectively.
Let us introduce SMP


t

transition probabilities:






012
,,0 ,
.
ii
teEPt Ee
eEEEE
 
 
As it is known [9], under the condition of the unique
EMC


,0
nn
  of SMP

t

stationary distribution
existence the following ratios take place
  
1
lim,,dd, 0,2,
i
i
tEE
teEme eme ei 



 



(8)
where


me is mean SMP

t

dwelling time in the
state eE

.
With the help of Formulas (5)-(7) integrals contained
in the Formula (8) are converted into:

A. I. PESCHANSKY 407
  
1
21 2121
000
ddddd,
x
g
E x
meeFtGttxGtthyxfyy M

 

 

  
 
2
21 100 000
21
00 00
ddd dd
dddd,
xx
nnn
n
E
x
gn
meeqlxxt tdxxyl y yGt t
q
xGtthyyxy sl ssM
p
 















 
 
 


0
21 10
21 1
00
dd
1dd ,
n
nn
n
E
l n
gg
n
p
meeqlxxx
q
q
MftHttqtHttMM
p

 











 









 


21 1
00
d1 dd.
n
l
n
gg
n
E
meeMf tHttqtHtt  





 






Consequently, the final probability that the server is free is defined by the formula

 


00
1
00
lim,,1
1d d
n
tl
n
gg
n
q
MM
p
pteE
M
ftH ttqtHtt







 









(9)
and the final probabilities p1 and p2 that the server is busy either with the primary service or with the re-servicing are:

 


11
1
00
lim,,,
1d d
n
tl
n
gg
n
M
pteE
M
ftH t tqtHtt
















(10)

 


22
1
00
lim, ,.
1d d
n
tl
n
gg
n
qM
p
pteE
M
ftH ttqtHtt
















(11)
It is necessary to note that the ratio
 


1
00
dd
n
l
n
gg
n
f
tH ttqtHt t 





determines an average value of requests lost per time unit
of a complete request service.
Let us define mean stationary dwelling times


i
TE
in the extracted subsets ,02
i
Ei.. According to [7 ] we
have
   
1
\
dd,,0.2
ii
ii
EEE
TEme eePeEi 






 .
(12)
One can define the values of expressions in denomi-
nators of Formulas (12). To do it let us take integrals of
both parts of the system (1) equations. The result is
     
    

0
021
\0000
21 21
0000
21 2
0
d,d21d12d 22d
dd12d12d
12d
EE
ePeE pftGttpGxxxpxxxpGxxx
pftGttpFxgxxpxxxpxxx
pp xxp

 
 
 


 
 
 


12121
,
q
pp  

A. I. PESCHANSKY 408
 
1
121
\0
d,10d,
EE
ePeEx x





   
2
22121 2121
\0000
d,d21ddd .
EE
ePeEq ftGttqGxxxq ftGttq ftGttq


 
 

Consequently, mean system dwelling times in the extracted subsets are defined by the formulas
  


01
00
1d,
n
l
n
gg
n
q
TEMftHtdtqtHt tMM
p






 




 
12
1
,.TE MTEM
p

  (13)
It is necessary to note that in case if with the probabil-
ity equal to 1 satisfactory request service is carried out
the system characteristics defined by the Formulas (9),
(10) and (13) coincide with ones found in [6].
To illustrate some subcases of the results gained let us
write down QS 10, 10,10MMMGGIM cha-
racteristics.
The system 10MM stationary characteristics. If
the RV ,,

 densities have the form

t
f
te



,
 
,,0
tt
gtete t



 sys tem stat ionary cha-
racteristics are defined by the formulas
11
01
1
2
1,1,
1,
qq
pp
pp
pp
pqq
 
 





 






  
012
11 1
,, . TETETE p



The system 10MG stationary characteristics. In
case when

,0,
t
gte t




and densities


,
f
t


t

are of general kind we have


,
t
n
lx e






,
n
l
g
H
xx


 


1
00
1
dd
.
n
l
n
gg
n
n
n
f
tH ttqtHt t
q
MMqMM
p


 










The Formulas (9)-(11) and (13) for defining system
stationary characteristics convert into
01
2
1,,
11
,
1
M
pp
qq
MMMM
pp
qM
p
pq
MM
p


 

 

 


  
01 2
11
,, .TETEMTEM
p



 
The system 10GI M stationary characteristics. For
this system the incoming flux of requests is generated by
RV

with density


g
t of a general kind, and




,,0
tt
ftete t




. Let us find the func-
tionals contained in the formulas for defining stationary
characteristics. We have
 
 

00
0
dd
d,
1
gg
tg
f
tH ttFtht t
g
ehtt g











where
 
0
d
t
g
gtet



  is Laplace transformation
of the density


g
t;


 



  
11
00
11
0
dd
1
d.
1
nn
n
ll
nn
gg
nn
l
nt n
gn
nn
qtHttq thtt
qehttql
g



















From the recurrence formula
 

1
0
1
00
d
d,d
nn
ng
ltsl sts
luuvytyuy










the recurrence formula for defining

n
l

 can be con-
cluded:
 

 
 
11
000
11
dd,d
d
d1
ty
t
nnn g
nn
ltltetltvytey
g
ll
g


 










 





That is why the sum of series

A. I. PESCHANSKY 409
  
1
1
1
nn
n
Sql
g






 is the solution of differential
equation
 
 
1
1
ql
SqS
g








.
This solution obeys the condition
  
0
lim 1q gS p



. This equation solution is defi-
ned by the formula
 

11
0
d
1
p
q
qxlx
Sx
gx






. As
  

11
gxg
lx xg







 we find out that
 

 
 

1
0
d
11
p
q
qxgxg
Sx
gxgx












.
Consequently, system stationary characteristics are
defined by the following ratios
 
 
  
0
1
0
11
1,
1d
1
p
q
q
qg
p
p
xgxg
M
x
xgx




 








 
















 
  
1
1
0
1,
1d
1
p
q
q
g
p
xgxg
M
x
xgx



 





 















 
 
 

2
1
0
1
,
1d
1
p
q
q
qg
p
p
xgxg
M
x
xgx



 





















 
12
11
,,TETEp




 
 
 

1
00
11
d.
11 1
p
q
qxgxg q
TE Mx
gg p xgx




 


  

 

 



 
5. Conclusions
In the present work semi-Markovian model of the opera-
tion of the single-server queue system 10GI G with
losses in which unsatisfactory request service quality is
admitted has been built. Service quality control is carried
out immediately, and in case of unsatisfactory service
quality re-servicing is executed until the service is regar-
ded as satisfactory. With the help of this model stationary
characteristics in explicit form have been defined. These
are: the final probabilities of system’s dwelling in a stan-
dby state, in the states of primary service and re-servic-
ing, and, besides, mean stationary dwelling times in these
states. These characteristics depend on mean time period s
between requests’ income, mean service time, average
value of requests lost per time unit of a complete request
service and the probability of unsatisfactory service. In
case of satisfactory service only the ch aracteristics found
coincide with the formerly defined ones.
6. References
[1] J. Riordan, “Stochastic Service Systems,” Member Tech-
nical Staff Bell Telephone Laboratories, Inc., John Wiley
& Sons, Inc., New York, 1962.
[2] B. V. Gnedenko and I. N. Kovalenko, “An Introduction
to Queuing Theory,” Nauka, Moscow, 1987.
[3] V. V. Anisimov, O. K. Zakusilov and V. S. Donchenko,
“Elements of Queuing Theory and System Asymptotic
Analysis,” Higher School, Kiev, 1987.
[4] B. V. Gnedenko and D. Konig, “Handbuch der Bedie-
nungsteorie II (For meen und Andere Ergebnisse),” Verlag,
Berlin, 1984.
[5] Y. Y. Obzherin and A. I. Peschansky, “Stationary Cha-
racteristics of a Single-Server Queue System with a Sin-
gle Waiting Space,” Cybernetics and System Analysis,
Vol. 42, No. 5, 2006, pp. 51-62.
[6] A. N. Korlat, V. N. Kuznetsov, M. I. Novikov and A. F.
Turbin, “Semi-Markovian Models of Restorable and Ser-
vice Systems,” Shtiintsa, Kishinev, 1991.
[7] V. S. Korolyuk and A. F. Turbin, “Markovian Restoration
Processes in the Problems of Sy stem Reliability,” Naukova
Dumka, Kiev, 1982.
[8] F. Beichelt and P Franken, “Zuverlässigkeit und Instan-
phaltung. Mathematische Methoden,” VEB Verlag Technik,
Berlin, 1983.
[9] I. N. Kovalenko, N. Y. Kuznetsov and V. M. Shurenkov,
“Stochastic Processes. Guidance,” Naukova Dumka, Kiev,
1983.