UDC 004.4
GPSS simulation model of a system with a general renovation
threshold based mechanism
Viana C. C. Hilquias∗ , I. S. Zaryadov∗† , S.I. Matyushenko??
∗
Department of Applied Probability and Informatics, Peoples’ Friendship University of
Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian
Federation
†
Institute of Informatics Problems, Federal Research Center “Computer Science and
Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333,
Russian Federation
Email: hilvianamat1@gmail.com, zaryadov-is@rudn.ru
For a queuing system with a general renovation with threshold control mechanism, a simulation model was built in the GPSS environment, which allows for various combinations of
generalized renovation and service mechanisms (service and renovation in the order in which
requests are received, service and renovation in the inverse order, service in the order of receipt - renovation in the inverse order, and vice versa) evaluate the main characteristics of
the system under study.
Key words and phrases: general renovation, threshold based control mechanism, GPSS,
simulation model.
1.
2.
Introduction
System description
Consider the M/M/1/∞ queuing system, shown in the Fig. 1, with the implemented
renovation mechanism and a threshold value Q1 .
µ
λ
Service
device
Queue
Q1
Figure 1. Queuing system model
Renovation mechanism: - if i ( number of requests in the queue ) ≤ Q1 , then
the renovation mechanism is not enabled, i.e. each serviced request simply leaves the
system. - if i > Q1 , then the renovation mechanism is activated, i.e. at the end of the
service, a request on the server can drop with probability q(k) q ≥ 0 exactly k requests
from the queue (q(0)pp < 1).
3.
pji
Transition probabilities
— transition probability of an embedded Markov chain — the probability that
a request entering the system will find exactly j, (j ≥ 0) requests in it, if the previous
one has found exactly i(i ≥ 0) requests.
πm (k) — the probability that as a result of serving exactly m requests, exactly k
requests will leave the system.
π0 (0) ≡ 1, π0 (k) ≡ 0, if k > 0
π1 (k) = q(k − 1),k ≥ 1
πm (k) =
k−1
X
πm−l (l)π1 (k − l) =
k−m+1
X
πm−1 (k − l)π1 (k − l), m ≥ 2, k ≥ m
l=1
l=m−1
pi,i+1 = P { between successive receipts, the request on the device did not complete
the service - nothing left the system}
Z ∞
=
e−µx dA(x) = α(µ)
0
where α(s)— Laplace–Stieltjes transform of the time distribution function between
successive arrivals of requests A(x).
If 0 < i ≤ Q1 — threshold value not exceeded
pi,j = P { between successive arrivals of a request into the system, i + 1 − j requests
left the system only due to service}
Z ∞
(µx)i+1−j −µx
=
e
dA(x) =
(i + 1 − j)!
0
(µx)i+1−j −µx
e
— probability of being serviced in time x exactly i + 1 − j requests.
(i + 1 − j)!
(−µ)i+1−j (i+1−j)
α
(µ),
=
(i + 1 − j)!
where α(k) (s)k-th derivative of Laplace–Stieltjes transform α(s)
If i ≥ Q1 + 1 (threshold Q1 in the queue is reached - the renovation mechanism is
enabled)
Pi,j = P { exactly i + 1 − j requests will leave the system both due to servicing and
updating } =
=
Z∞i+1+j
X
0
=
πm (i + 1 − j)
(µx)m −µx
e
dA(x) =
m!
πm (i + 1 − j)
(−µ)m
· α(m) (µ)
m!
m=1
i+1+j
X
m=1
Pi,0 = 1 −
i+k
X
Pi,j
j=1
4.
Stationary probability distribution
Let πi be the stationary probability, along the nested Markov chain, that a request
entering the system will find exactly i other requests in it. Then the system of equations
for the probabilities πi has the form:
π0 =
∞
X
i=0
πi · pi,0
πi =
∞
X
πk · pk,i , i ≥ 1
k=i−1
Let’s assume that at i ≥ Q1 + 1πi = πQ1 +1 · g i−Q1 −1 ⇔ πQ1 +1+i = πQ1 +1 g i , i ≥ 0.
πi =
∞
X
πk pk,i , i ≥ 1,
k=i−1
If i ≥ Q1 + 1, then
πi = πQ1 +1 ġ i−Q1 −1
Let 1 ≤ i ≤ Q1 , then
πi =
Q1
X
πi pk,i + πQ1 +1 · g i−Q1 −2 ∗
k=i−1

∗ g − α(µ) −
Q1X
+1−i
m=1
Let i = Q1 , then
(−µg) (m)
α
(µ)
m!
Q1 +1−i−m
X

πm (m + k)g
k
k=0
πQ1 = πQ1 +1 · g −1 .
5.
Simulation modeling in GPSS
Below is the result of system simulation in GPSS. System parameters during simu1
1
, λ = and the threshold value Q1 = 2.
6
5
GPSS World Simulation Report
lation: 10000000 requests, µ =
Tuesday, February 28, 2023 12:47:00
START TIME
0.000
NAME
ADEL
ALAM
AMU
KDEL
KSERV
NAK
NEXT1
NEXT2
NEXT3
NEXT4
NEXT5
PDEL
POROG
PRIB
PSERV
QUE
END TIME BLOCKS FACILITIES STORAGES
1999985.705
28
1
0
VALUE
10003.000
10000.000
10001.000
10005.000
10004.000
10014.000
7.000
22.000
15.000
21.000
23.000
10007.000
10002.000
10013.000
10006.000
10012.000
SYST
TDEL
TSERV
WORK
10011.000
10010.000
10009.000
10008.000
LABEL
1
GENERATE
2
MARK
3
QUEUE
4
QUEUE
5
TEST
6
LINK
NEXT1
8
DEPART
9
TABULATE
10
ADVANCE
11
RELEASE
12
DEPART
13
TEST
14
UNLINK
NEXT3
16
SAVEVALUE
17
TEST
18
SAVEVALUE
19
TEST
20
SAVEVALUE
NEXT4
NEXT2
NEXT5
24
DEPART
25
TABULATE
26
SAVEVALUE
27
SAVEVALUE
28
TERMINATE
FACILITY
PRIB
QUEUE
SYST
QUE
TABLE
TSERV
_ 0.000
0.300
0.600
0.900
1.200
1.500
1.800
2.100
2.400
TDEL
0.000
ENTRIES
8761418
LOC BLOCK TYPE
10000000
10000000
10000000
10000000
10000000
7303937
7
SEIZE
8761418
8761418
8761418
8761418
8761418
8761418
6065355
15
SAVEVALUE
8761418
8761418
1251417
1251417
11349
21
UNLINK
22
TERMINATE
23
DEPART
1238582
1238582
1238582
1238582
1238582
ENTRY COUNT CURRENT COUNT RETRY
0
0
0
0
0
0
0
0
0
0
0
0
8761418
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8761418
0
0
0
0
0
0
0
0
0
0
0
0
1251417
0
0
8761418
0
0
1238582
0
0
0
0
0
0
0
0
0
0
0
0
UTIL.
AVE. TIME AVAIL. OWNER PEND INTER RETRY DELAY
0.730
0.167 1
0
0
0
0
0
MAX CONT. ENTRY ENTRY(0) AVE.CONT. AVE.TIME
AVE.(-0) RETRY
28
0 10000000
0
1.983
0.397
0.397
0
27
0 10000000 2696063
1.253
0.251
0.343
0
MEAN
STD.DEV.
RANGE
0.223
0.275
0.000
2696063
30.77
0.300
3486888
70.57
0.600
1725959
90.27
0.900
591216
97.02
1.200
181426
99.09
1.500
54749
99.71
1.800
17136
99.91
2.100
5403
99.97
2.400
1763
99.99
- _
815
100.00
0.448
0.310
0.300
462847
37.37
RETRY FREQUENCY CUM.%
0
0
0.300
0.600
0.900
1.200
1.500
1.800
2.100
2.400
-
0.600
0.900
1.200
1.500
1.800
2.100
2.400
_
USER CHAIN
NAK
SAVEVALUE
ALAM
AMU
POROG
ADEL
KSERV
KDEL
PSERV
PDEL
WORK
FEC XN
PRI
10000001
0
468260
202460
70760
23204
7475
2406
800
370
SIZE RETRY
0
0
75.18
91.52
97.23
99.11
99.71
99.91
99.97
100.00
AVE.CONT
1.253
ENTRIES
7303937
MAX
27
AVE.TIME
0.343
RETRY
VALUE
0
5.000
0
6.000
0
2.000
0
1.000
0
8761418.000
0
1238582.000
0
0.876
0
0.124
0
2.000
BDT
ASSEM CURRENT
1999985.716
10000001
NEXT
0
PARAMETER
1
VALUE
The figure 2-3 below shows the diagrams of the waiting time for the start of service
and the time spent in the system by a dropped request.
Figure 2. Wainting time in the queue
Figure 3. Wainting time in the queue
Below in the table 1 is the result of system simulation in GPSS for differents threshold
1
1
,λ= .
6
5
values. System parameters during simulation: 1000000 requests, µ =
Table 1
GPSS simulation for differents threshold values
Threshold value
2
Probability of servicing tasks
Probability of dropping tasks
Serviced tasks
Dropped tasks
Average wait time
to start service
Average time in the
queue of dropped applications
Average queue length
Maximum queue length
0.877
0.123
876566
123434
5
10
20
30
40
50
0.943 0.98 0.997 0.9996 0.99997
1
0.057 0.02 0.003 0.00034 0.00003
0
942815 980203 997001 999619 999974 1000000
57185 19797 2999
381
26
0
0.251
0.394
0.585
0.765
0.793
0.773
0.75
0.343
0.502
0.716
0.92
0.955
0.928
0
1.253
22
1.969
23
2.923
28
3.819
34
3.968
41
3.859
45
3.7
47
Below in the table 2 is the result of system simulation in GPSS for differents service
1
and the threshold
5
value Q1 = 2.
rates. System parameters during simulation: 1000000 requests, λ =
Table 2
GPSS simulation for differents service rates
Service rate
Probability of servicing tasks
Probability of dropping tasks
Serviced tasks
Dropped tasks
Average wait time
to start service
Average time in the
queue of dropped applications
Average queue length
Maximum queue length
1/s
2/s
4/s
0.5
0.5
0.72
0.5
0.5
0.28
499784 499788 719659
500216 500212 280341
6/s
10/s
0.877
0.123
876566
123434
0.971
0.029
970538
29462
20/s
50/s
0.997 0.99991
0.003 0.00009
997329 999910
2671
90
149935 24937
0.66
0.251
0.078
0.016
0.002
149935 24937
0.734
0.343
0.161
0.065
0.022
749674 124504 3.297
1501586 248961 42
1.253
22
0.391
16
0.08
8
0.011
6
6.
Conclusion
In this paper, for a queuing system with a threshold update mechanism a simulation
model was built in the GPSS environment, which allows for various combinations of
generalized renovation and service mechanisms (service and renovation in the order in
which requests are received, service and renovation in the inverse order, service in the
order of receipt - renovation in the inverse order, and vice versa) evaluate the main
characteristics of the system under study.
References
1.
2.
3.
4.
5.
F. Baker,G. Fairhurst, IETF Recommendations Regarding Active Queue Management, RFC 7567, Internet Engineering Task Force. https://tools.ietf.org/
html/rfc7567
S. Floyd, V. Jacobson, Random Early Detection Gateways for Congestion Avoidance, IEEE/ACM Transactions on Networking, 4 (1), 397–413, 1993.
K. Ramakrishnan, S. Floyd, D. Black, The Addition of Explicit Congestion Notification (ECN) to IP. RFC 3168, Internet Engineering Task Force, 2001. https:
//tools.ietf.org/html/rfc3168
I. S. Zaryadov, A. V. Pechinkin, Stationary Time Characteristics of the GI/M/n/∞
System with Some Variants of the Generalized Renovation Discipline, Automation
and Remote Control, 70 (12), 2085–2097, 2009. doi:10.1134/S0005117909120157.
C. C. Hilquias Viana, I. S. Zaryadov, T. A. Milovanova, V. V. Tsurlukov, A. V. Korolkova, D. S. Kulyabov, The General Renovation as the Active Queue Management
Mechanism. Some Aspects and Results, in: Communications in Computer and Information Science, vol. 1141, 488-502, 2019. doi:10.1007/978-3-030-36625-4_39.
УДК 004.4
Моделирование системы массового обслуживания с одним
порогом и обновлением в GPSS
Виана К.К. Илкиаш∗ , И. С. Зарядов∗† , С. И. Матюшенко
∗
Кафедра прикладной информатики и теории вероятностей,
Российский университет дружбы народов,
ул. Миклухо-Маклая, д.6, Москва, Россия, 117198
†
Институт проблем информатики, , ул. Вавилова, д. 44, кор. 2, Москва, Россия
Email: hilvianamat1@gmail.com, zaryadov-is@rudn.ru
Для системы массового обслуживания с обобщенным обновлением с пороговым механизмом обновления построена имитационная модель в среде GPSS, которая допускает
различные комбинации механизмов обобщенного обновления и обслуживания (обслуживание и обновление в порядке поступления заявок, обслуживание и обновление в
обратном порядке, обслуживание в порядке поступления - сброс в обратном порядке и
наоборот) оценивают основные характеристики исследуемой системы.
Ключевые слова: обобщенное обновление, пороговый механизм управления, GPSS,
имитационная модель.