Queueing systems with different types of renovation mechanism and thresholds as the mathematical models of active queue management mechanism

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Introduction
In modern communication networks the problem of congestion avoidance does not have a satisfying solution [1] and the development of active queue management (AQM) algorithms appears to be the actual task for researches and practitioners.
A numerous number of AQM schemes have been proposed [2], some of them were investigated and standardised by IETF working group on "Active Queue Management and Packet Scheduling" [3].
For the most AQM algorithms models the performance analysis is performed by simulation (for example, [2]) and that is why the bridges between the available use-case results and the available analytic results are very few (see, for example, [4]- [7]).
In this paper the mathematical models of RED-like ( 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 length [8]- [10], in our models the decision about a possible packet drop is made at the momemnts of service completions) algorithms with renovation and one or several thresholds (which determine not only the place in the buffer from which the packets are dropped, but also the place to which the reset of packets occurs).
The structure of the article is following. The section 2 gives the brief description active queue algorithms, especially of the classic RED algorithm. The section 3 consists of foyr subsections: in 3.1 the definitions of different types of renovation are given as well as a brief overview of queueing systems with renovation; in 3.2 the models with renovation and a single thresholds are described; the subsectionn 3.3 is devoted to models with renovation and two thresholds (for one of these models some analitycal results are presented); the last subsection 3.3 describes renovation models with three thresholds. In section 4 the comparison of results for RED algorithm and some renovation models is presented, based on experimental results from [11]- [13]. These results confirm assumptions that the use of the renovation mechanism in the single server queues under heavy overload conditions and some other constraints allows one to achieve at least the same performance level as by the classical random early detection algorithm. The last section 5 concludes the paper with the short discussion.

Active queue management algorithms. The brief description of RED algorithm module
According to RFC 7567 [1] active queue management (AQM) is considered as a best practice of network congestion avoidance (reducing) in Internet routers. The active queue management is the policy of dropping packets inside a buffer associated with a network interface controller (NIC) before that buffer becomes full (or gets close to becoming full) and this policy is based on some rules (algorithms ) such as: -Random Early Detection (random early discard or random early drop) (RED) [8]- [10], [14] -is a queuing discipline for a network scheduler suited for congestion avoidance by pre-emptively dropping packets before the buffer becomes completely full based on predictive models to decide which packets to drop; -Explicit Congestion Notification (ECN) [9], [15] and its modifications [16]- [19] -allows end-to-end notification of network congestion without dropping packets (opposed to RED); -controlled delay (CoDel) [20] and its modifications [21], [22] -a scheduling algorithm for the network scheduler, designed to overcome bufferbloat in networking hardware by setting limits on the delay network packets experience as they pass through buffers in this equipment; -BLUE [23] and its modifications [24]- [26] -operates by randomly dropping or marking packet with explicit congestion notification mark before the transmit buffer of the network interface controller overflows and it requires little or no tuning to be performed by the network administrator; -CAKE (Common Applications Kept Enhanced) [27], [28] -is a shapingcapable queue discipline which uses both AQM and FQ, itcombines COBALT, which is an AQM algorithm combining Codel and BLUE Early AQM disciplines (notably RED and SRED) require careful tuning of their parameters in order to provide good performance, but modern AQM disciplines (Blue, CoDel, CAKE and new modifications of RED -the overview may be seen in [29], [30]) are self-tuning, so they can be run with their default parameters.
The classic RED [8] is an active queue management algorithm with two thresholds ( and ), which monitors the average queue size and drops packets (or marks packets when it used in conjunction with ECN) based on statistical probabilities () [8]: where max is the maximum of dropped probability (probability of marking), and̂is the queue length control parameter (low-pass filter) calculated by formula:̂+ is a weight coefficient of the exponentially weighted moving-average.

The definition of renovation mechanism
The renovation mechanism was introduced in [31]: at the moment of the end of its service the packet on the server may either just leave the system with some non-zero probability , or may empty the buffer with the renovation 308 DCM&ACS. 2020, 28 (4) 305-318 probability = 1 − . In [31] the steady-state probability distributions for several types of queueing systems were presented.
Later the model of renovation with feedback (or repeated service) was introduced in 2007 by professor P. P. Bocharov in [32]. The renovation with feedback means that after emptying the buffer with probability the served packet starts another round of service. The main characteristics in matrixanalytical form were obtained. Another system with renovation and feedback was studied in [33], [34], where the retrial queueing system with renovation and recurrent input flow was investigated.
The generalisation of renovation mechanism was proposed in 2008 by professor A. V. Pechinkin [35]: at the moment of the end of service the packet may drop from the buffer with probability ( ) exactly ( ⩾ 1) other packets and leaves the system or may just just leave the system without any effect on it with the complementary probability = 1 − ∑ =1 ( ).
In [12], [13], [35]- [39] queueing systems with different types of renovation and service disciplines were presented. It was proved that for / / /∞ systems with general renovation (or just renovation) the steady-state probability distribution has the geometric form and sojourn time distribution of a served (lossed) packet has an exponential form.
The study of queueing systems with renovation mechanism and thresholds was started in [12], [13].

The mathematical model with renovation and one threshold
The simplest mathematical models of renovation mechanism with thresholds are the models with one threshold 1 for queue length.
If the current queue lengtĥis larger than 1 (̂> 1 ) then a packet on the server at the moment of the end of its service may either leave the system with probability or may drop a packet (packets) from the buffer and leave (renovation mechanism).
The next types of renovation may be considered: -the "simple" renovation of the first type -the fixed size group of ( ⩾ 1) packets will be dropped with fixed probability ; -the "simple" renovation of the second type -the fixed size group of ( ⩾ 1) packets will be dropped with fixed probability (but 1 will remain anyhow) -the "complete" ("full") renovation (or just renovation) -all packets from the buffer will be dropped with probability = 1 − , the system will be empty; -the "incomplete" renovation -all packets (that are after the threshold) willbe dropped with probability = 1 − , 1 packets will remain in the system; -the general renovation of the first type -exactly ( ⩾ 1) packets from the buffer will be dropped with probability ( ) if in the buffer there are more then packets or the system will be empty if in the buffer were were less (or equal) then packets; -the general renovation of the second type -exactly ( ⩾ 1) packets from the buffer may be dropped with probability ( ) if there were more then + 1 packets in the buffer, or 1 packets will remain in the system if in the buffer were less (or equal) then + 1 packets.
For the last two cases of general renovation the following condition on probabilies and ( ) ( ⩾ 1) must be met: is the probability, that a served packet will just leave the system without dropping other packets.
The minus of the first, the third and the fifth drop mechanism is that too many packets may be dropped, the minus of the second, the fourth and the sixth drop mechanism is that the buffer may remain overflowed. Models with two thresholds allow to get rid of these minuses.

The mathematical model with renovation and two thresholds
In this part of the article we will discuss models of RED-like algorithm based on queueing systems with renovation and two thresholds.
The thresholds of the first type: the first threshold 1 determines when packets start to be dropped, the second threshold 2 ( 2 > 1 ) sets the maximum value of drop probability.
The types of renovation policy. At the moment of the end of a packet service the current queue lengtĥis compared with thresholds and if̂⩽ 1 then no one of the packets from the buffer is dropped.
-If 1 + 1 ⩽̂⩽ 2 then the fixed size group of ( ⩾ 1) packets will be dropped with fixed probability . If̂⩾ 2 + 1 then the fixed size group of packets in the buffer will be dropped with maximal probability max ( < max ⩽ 1. -If 1 + 1 ⩽̂⩽ 2 then the fixed size group of packets will be dropped with probability (̂), 0 < (̂) < max . If̂⩾ 2 + 1 then the fixed size group of packets in the buffer will be dropped with maximal probability max (this model was considered in [12] for = 1 (the last packet is dropped) and the steady-state probability distribution of packets in the system(for imbedded Markov chain) as well as some probabilistic (the probability that the arriving packet will be dropped or served) and sojourn time characteristics for served packets were obtained). -If 1 + 1 ⩽̂⩽ 2 then the fixed size group of packets will be dropped with fixed probability . If̂⩾ 2 + 1 then all packets except the first 1 will be dropped with probability . -If 1 + 1 ⩽̂⩽ 1 then the fixed size group of packets will be dropped with probability (̂). If̂⩾ 2 + 1 then all packets except the first 1 will be dropped with probability . -If 1 + 1 ⩽̂⩽ 2 then the arbitrary size group of packets will be dropped with given probability ( ) (0 < ( ) < 1). If̂⩾ 2 + 1 then all packets except the first 1 will be dropped with probability . For all types of renovation policy the following condition on probabilies , (̂), ( ) , max , ( ⩾ 1) must be met: where , (̂) ( ⩾ 1) and are the probabilities that the served packet will leave the system without dropping any other packets from the buffer (if 1 + 1 ⩽̂⩽ 2 ), and (2) max and (2) are the probabilities that the served packet will leave the system without dropping any other packets from the buffer (if̂⩾ 2 + 1).
The thresholds of the second type: the first threshold 1 determines when packets start to be dropped, the second threshold 2 determines the place to which packets will be dropped from the queue ( 1 ⩾ 2 ). If the current queue sizêis less than 1 , then a served packet just leave the system. But if̂⩾ 1 + 1 then the following types of renovation (dropping) mechanism may be applied: -the fixed size group of ( ⩾ 1) packets will be dropped with fixed probability (anyhow 2 packets will remain in system); -with probability only 2 packets will remain in system, all others packets will be dropped; -the arbitrary size group of packets will be dropped with given probability ( ) (0 < ( ) < 1) (anyhow 2 packets will remain in system).
For all types of renovation policy the following condition on probabilies , (̂) ( ⩾ 1) and must be met: where ( ⩾ 1), and (2) are the probabilities that the served packet will leave the system without dropping any other packets from the buffer (if 1 + 1 ⩽̂). The minus of these policies is that the drop probability ceases to depend on the queue length and it is impossible to set the maximum drop probability. Models with three thresholds allow to get rid of this minus.
For the third model in [13] the steady-state probability distribution of packets ( ⩾ 0) in the system(for imbedded Markov chain), some probabilities were obtained and represented by geometric form (when the threshold 1 is overcomed): where is the unique solution of the equation = ( (1 − ( ))), and belongs to interval (0; 1), ( ) is the Laplase-Stieltjes transformation of interrarrival time distribution function ( ), ( ) is the probability generating function for probabilities ( , ) that packets will be served and packets will be dropped from the buffer, Also the probability that the arriving packet will be dropped and sojourn time characteristics for dropped packets were obtained in form of integral equations.

The mathematical model with renovation and tree thresholds
In this part of the article models with renovation and three thresholds ( 1 , 2 and 3 ) will be formulated. The first threshold 1 determines when packets start to be dropped, the second threshold 2 ( 2 > 1 ) sets the maximum value of drop probability, the third threshold 3 ( 3 < 1 ) determines the place to which packets may be dropped from the queue.
The following types of renovation mechanism may be defined. At the moment of the end of a packet service the current queue lengtĥis compared with thresholds and if̂⩽ 1 then no one of the packets from the buffer is dropped.
-If 1 + 1 ⩽̂⩽ 2 then the fixed size group of ( ⩾ 1) packets will be dropped with fixed probability . If̂⩾ 2 + 1 then the fixed size group of packets in the buffer will be dropped with maximal probability max ( < max ⩽ 1, but 3 packeta anyhow will remain in the system. -If 1 + 1 ⩽̂⩽ 2 then the fixed size group of packets will be dropped with probability (̂), 0 < (̂) < max . If̂⩾ 2 + 1 then the fixed size group of packets in the buffer will be dropped with maximal probability max , but 3 packeta anyhow will remain in the system. -If 1 + 1 ⩽̂⩽ 2 then the fixed size group of packets will be dropped with fixed probability . If̂⩾ 2 + 1 then all packets except the first 1 will be dropped with probability . -If 1 + 1 ⩽̂⩽ 2 then the fixed size group of packets will be dropped with probability (̂). If̂⩾ 2 + 1 then all packets except the first 1 will be dropped with probability . -If 1 + 1 ⩽̂⩽ 2 then the arbitrary size group of packets will be dropped with given probability ( ) (0 < ( ) < 1). If̂⩾ 2 + 1 then all packets except the first 1 will be dropped with probability . For all types of renovation policy the following condition on probabilies 1 , , (̂), (̂), ( ) , max , ( ⩾ 1) must be met: where , (̂) ( ⩾ 1) and are the probabilities that the served packet will leave the system without dropping any other packets from the buffer (if 1 + 1 ⩽̂⩽ 2 ), and max and (2) are the probabilities that the served packet will leave the system without dropping any other packets from the buffer (if̂⩾ 2 + 1).

The comparison of some renovation models with RED algorithm
In this section we will compare values of the probability of packet being dropped from the system loss for RED and TailDrop algorithms [4], [6], [7] and values of the probability loss , obtained by formulas derived for queueing system with general renovation [11], queueing system with general renovation and feedback [33], [34] and queueing system with general renovation and two thresholds [12].
As can be see from the table 1, according to the values of the loss , renovation mechanism can perform as good as RED in the wide range of the offered load . Table 1 The value of the loss probability for Taildrop, RED, general renovation, general renovation with feedback, general renovation with two thresholds Loss probability Taildrop RED renov. ren-fd. ren-thr.

Conclusion
The main task of the authors in this work was to formulate different types of models with various renovation policies and one or more threshold values in a buffer of a system. These thresholds allows to control the drop probability of renovation mechanism. Some of the described models have already been researched or are in the process of research (links to scientific publications are provided) The presented numerical experiments show that the results remain qualitatively the same for RED-type AQM with other dropping functions. Being defined by parameters, the renovation mechanism is very flexible and this constitutes its strength and weakness. By varying the values of the renovation probabilities ( ), it is possible to carry out conditional optimisation, but good searching procedures are required here.