Analysis of queuing systems with threshold renovation mechanism and inverse service discipline

. The paper presents a study of three queuing systems with a threshold renovation mechanism and an inverse service discipline. In the model of the first type, the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims). In the second model, the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In the model of the third type (generalizing the previous two models), two threshold values are used: one to activate the mechanism for dropping requests, the second — to set a safe zone in the queue. Based on the results obtained earlier, the main time-probabilistic characteristics of these models are presented. With the help of simulation modeling, the analysis and comparison of the behavior of the considered models were carried out.


Introduction
According to [1] the problem of congestion avoidance for communication networks does not have a satisfying solution, so the development and the analysis of new active queue management (AQM) algorithms appears to be the actual task for researches [2]- [13] and practitioners [14]- [24].
In this paper we will consider queuing systems with probabilistic renovation mechanism, which allows to adjust the number of packets in the system by dropping (resetting) them from the queue depending on the ratio of a certain control parameter with specified thresholds [25], [26] at the moment of the end of service on the device (server) [27]- [29] in contrast to standard RED algorithm, when a possible reset occurs at the time of the next packet arrival and the control parameter is an exponentially weighted average queue

The first model
Consider the / /1/∞ queuing system, shown in the figure 1, with the implemented renovation mechanism and a threshold value 1 , which determines the boundary in the queue, starting from which the dropping of customers begins. If the current number of packets in the system ⩽ 1 + 1 (the threshold value 1 is not been overcome), then none of the packets will be dropped from the queue. If the current number of packets in the system ⩾ 1 + 1, then with probability the packet finishing the service can drop all packets from the queue and leave the system, or with probability = 1 − the serviced packet simply leaves the system. Let (loss) be the probability that the packet received in the system will be dropped by renovation mechanism and let (loss) be the probability that a packet arriving and finding in the system exactly packets will be dropped.

Let
(loss) ( ) be the probability that in a time less than a packet that finds other packets in the system will be dropped. Then: ( ) is the probability that in time less than the packet, before which there are other packets in the queue and after which there are other packets, will be dropped, , ⩾ 0.

Time characteristics of the system
Let (serv) ( ) and (loss) ( ) be the distribution functions of the time spent in the queue by the served and dropped packets. ,0 ( )) ⋅

Time characteristics for a dropped packet
′ , we also will consider several cases.
a) The first case is when +1+ ⩽ 1 , so the selected packet can be dropped only due to the reception of new packets in the system and overcoming the threshold value (loss) , , − +1 ( − ).

The second model
The second queuing model is also / /1/∞ queuing system, shown in the figure 2, with the implemented renovation mechanism, but the threshold value 1 determines the boundary in the queue, starting from which the dropping of customers begins and also determines the safe zone from where packets cannot be dropped. If the current number of packets in the system is less or equal to 1 + 1 (the threshold value 1 has not been overcome), then none of the packets will be dropped from the queue. If the current number of packets in the system is greater then 1 + 1, then with probability the packet, finishing the service and leaving the system, will drop all packets from the queue (outside the safe zone), or with probability = 1 − the serviced packet simply leaves the system.

Let
be the steady-state probability distribution of the embedded Markov chain that the packet comming into the system will find in it other packets ( ⩾ 0) [37], [38].
Let (loss) and (serv) be the probability that the received packet in the system will be dropped from the queue or will be transferred to service device. The (serv) is the auxiliary probability that the packet will be served if it finds other packets in the system.

Time characteristics for serviced packets
(serv) ( ) is the cumulative waiting time distribution function for the accepted into the system packet, (serv) ( ) is the cumulative waiting time distribution function for the accepted into the system packet, if at the moment of its arrival there were other packets in the system. Then: The auxiliary functions (serv) , ′ ( , ⩾ 0) are the distribution functions and the densities of distribution functions of the time spent by the served packet in the queue, if there were other packets in the queue before the considered one and others after it. a) If = 0, then the cumulative distribution functions (serv) ( ) = 1, ( = 0). b) If 0 < ⩽ 1 -(the safe zone is not completely filled) then the received in the system packet will be in the safe zone (cannot be dropped). Then ,1 ( − ).
b.1) 0 < + ⩽ 1 , > 0 (taking into account the packets that came after ours), the threshold value 1 has not been overcome in the queue, that is, the renovation mechanism has not turned on. Then , − +1 ( − ).
b.2) 1 < + 1 ( > 0) the renovation mechanism was activated, but our packet is in a safe zone. Then c) ⩾ 1 + 1 -at the time of receipt of our packet, the safe zone is filled and there are packets outside the safe zone -the renovation mechanism is enabled. Then w (serv) , − +1 ( − ).

Time characteristics for dropped packets
Let (loss) ( ) be the cumulative distribution functions of the time spent by the packet in the queue before dropping.
(loss) ( ) is the conditional probability that in a time less than the packet that has found exactly of other packets in the system will be dropped from the queue. The auxiliary functions ′ ( , ⩾ 0) are the distribution functions and the densities of distribution functions of the time spent by the dropped packet in the system, if there were other packets in the queue before the considered one and others after it. a) 0 ⩽ ⩽ 1 (that is, the system was either empty, or at least there was one free space in the safe zone) , − − +1 ( ).

The third model
Consider the / /1/∞ queuing system, shown in the figure 3. In this section, a single-server queueing system with an infinite queue capacity and two threshold values is considered. Threshold values: -1 -the threshold value in the queue, when overcoming which by the queue length packets (from 1 + 1) will be dropped from the queue with a probability . -2 -the threshold value in the queue to which packets are dropped (i.e. packets standing in the queue up to the 2 threshold are not dropped).

The service probability and loss probability of the received packet
Let's introduce the probability (serv) that the packet, entering the system, will be served, auxiliary probabilities (serv) ( ⩾ 0) of incoming packet to be served if there were other ( ⩾ 0) packets in the system, and auxiliary probabilities (serv) , ( ) that during the time the packet, which found exactly other packets in the system at the moment of arrival and behind which there are more packets, will be served where -the stationary probabilities [37], [38]. Let's consider several cases a) The first one, when the system is empty: b) The second case is when 1 ⩽ ⩽ 2 , so (serv) = 1.
c.2) the second subcase, + 1 + > 1 + 1 -the 1 threshold in the queue has been overcome, so the renovation mechanism has been activated (serv) , , − +1 ( − )+ , − +1 ( − ). d) the fourth case is when the 1 threshold in the queue has been overcome at the moment of the arrival of the considered packet, ( > 1 ) so the renovation mechanism has been already activated (serv) , ,0 ( ) .

Loss probability of the received packet
where (loss) -the probability that the incoming packet will be dropped if at the moment of its arrival there were , ⩾ 0 other packets in the system, and b.2) +1+ > 1 +1 -(the 1 threshold was overcome due to applications after the incoming one) (loss) , , − +1 ( − )+ , − +1 ( − ).

Time characteristics of the system
Let (loss) ( ) and (serv) ( ) be the cumulative distribution functions of the time spent in the system by the packet before being dropped or served. The auxiliary functions 0,0 ( ) = 0. b) If the total number of packets in the system has not overcome the threshold 2 (0 < ⩽ 1 , + + 1 ⩽ 1 ), then the considered packet will be in the safe area and the renovation mechanism is not enabled.
c) The case, when at the moment of arrival of the considered packet there were 0 < < 2 other packets in the system (our packet was in the safe area), but currently the total number of packets in the system is equal to + + 1 > 1 (so the renovation mechanism is enabled) d) The case, when at the moment of arrival of the considered packet there were 2 < < 1 other packets in the system (our packet was out of the safe area), includes several subcases. d.1) The first subcase -currently the total number of packets in the system is 2 < + + 1 ⩽ 1 (the renovation mechanism is not enabled) w (serv) , , − +1 ( − ). d.

GPSS simulation results
Below (see table 1) is presented a table with GPSS simulation results that was performed with the following initial parameters: threshold value 1 = 30, arrival rate -14 task per 1 unit of time, service rate -16 task per 1 unit of time, and the simulation time is 100000 units of time) for different drop probabilities.
The table 2 shows the results of GPSS simulation that was performed with the following initial parameters: arrival rate -14 task per 1 unit of time, service rate -16 task per 1 unit of time, = 0.01, and the simulation time is 100000 units of time) for different threshold values. For the third model the threshold value 2 = 10.

Conclusion
Based on the simulation results 1, the following conclusions can be drawn. The largest number of dropped packets, as expected, is observed in the first model, the smallest -in the second model (due to the safe zone). The third model shows an average result compared to the first and the second models. The largest number of serviced packets is in the second model, then -in the third model. The smallest number of serviced packets is in the first model.
The probability of a packet to be dropped is about five times greater for the first model than for the second model, and 20-30 percent more than for the third model.
The average waiting time for the second model is about 5-10 percent greater than the same characteristic for the first and third models.
As the value of the renovation probability increases, the drop probability increases for all three models, and the service probability decreases accordingly. Also, with an increase of the renovation probability , both the average and maximum queue lengths decrease, and the average waiting time also decreases.
Based on the simulation results 2, the following conclusions can be drawn. With an increase of the threshold value 1 responsible for switching on the renovation mechanism, the number of dropped packets decreases for all three models (the second model is characterized by the smallest number of dropped packets), the service probability increases to unity (the second model), and the drop probability decreases almost to zero. The average and maximum queue lengths increase, and the values for the first and third models become approximately the same. The average waiting time also increases, and again for the first and third models, the values become approximately the same.
The third model, which generalizes the first and the second models, shows average results compared to the above models, and is more preferable for use as a queue length management model.