In my paper, Z.I. Flikop “Some Problems with a Design of Hierar=
chical
Management Systems” [in Russian], Mehanizatcia i avtomatizatcia ypravlenia, 1971 # 5, I have proposed the criterion for t=
he
evaluation efficiency of multiphase queuing systems that con=
sist
of different type of servers and provide service to different type of eleme=
nts
(customers). I do not have th=
is
paper, however I have a book A. L. =
Lifshitc, E.A. Maltc =
“Statistical Modeling of Queuing
System” [in Russian], Mos=
cow,
1971 that cited this crit=
erion
from my paper.
The criterion is:
![](3D"Objectivefunction_files/image002.gif")
where:
![](3D"Objectivefunction_files/image004.gif")
- is a cost=
(per
unit of time) associated with a waiting time in the queue of a customer of =
an l type;
k – is a number of customer types;
=
![](3D"Objectivefunction_files/image006.gif")
- is a numb=
er of l type customers, that currently a=
re
servicing by the system and also =
being in all queues of all phases of a system;
=
![](3D"Objectivefunction_files/image008.gif")
- is a
probability that m customers of=
an l type currently are servicing by =
all
phases of the system and also being
in all queues of all phases of the system;
=
![](3D"Objectivefunction_files/image010.gif")
- is a cost=
(per
unit of time) associated with idling of a j
type server;
&n=
bsp;
n – is a number of server ty=
pes;
&n=
bsp;
![](3D"Objectivefunction_files/image012.gif")
- is a cost=
(per
unit of time) associated with a servicing of a customer by a server of a j type;
&n=
bsp;
![](3D"Objectivefunction_files/image014.gif")
- is a numb=
er of
servers of j type;
&n=
bsp;
![](3D"Objectivefunction_files/image016.gif")
-is a numbe=
r of
idling servers of a j type;
&n=
bsp;
![](3D"Objectivefunction_files/image018.gif")
- is a probability that customer is being served by a j type server;
&n=
bsp;
![](3D"Objectivefunction_files/image020.gif")
- is a
probability that j type server =
is
idle;
&n=
bsp;
![](3D"Objectivefunction_files/image022.gif")
- is cost
associated with time that customers spend in the system;
&n=
bsp;
![](3D"Objectivefunction_files/image024.gif")
- is a cost associated with idling of servers;
![](3D"Objectivefunction_files/image026.gif")
- is a cost associated with usage of a servicing system.<=
/p>
In addition to the criterion, A.L. Lifshitc and E.A. Maltc in theirs book gave the fol=
lowing
explanation:
![](3D"Objectivefunction_files/image028.gif")
- is an av=
erage
number of customers of type l t=
hat
are in the system;
![](3D"Objectivefunction_files/image030.gif")
- is an ave=
rage
number of idling servers of type j<=
/i>;
![](3D"Objectivefunction_files/image032.gif")
- is an average numbers of working servers of type j;
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