Queueing Systems with Renovation

We study in this paper-single channel queueing systems with renovation. Such models are often encountered in computer science applications. Explicit solutions are discussed for the queues with Poisson input flow and for queues with general input and exponential service times.


Introduction
Consider a single-channel queueing system with FIFO discipline and the following modification.After each customer is serviced, all the others leave the queue with pro- bability q.Only a customer who has been serviced leaves the system with probability p 1q.The probability q, which we call renovation probability, is essential to our analysis.This probability defines many important features of our model.This model is a generalization of the classical single-server system.Below we pre- sent a short review of previous results concerning only the queues with general distri- bution of the input flow.The results related to such cases as Markovian input or deterministic input are well known and are described in many books and papers (see, e.g., monographs by Asmussen [1], Takagi [12], Klimov [9, 10] and Jaiswal [6]).The case of general input flow and exponentially distributed service times was considered in [6].It is shown that if N < ec, the Laplace transform of the stationary queue length distribution can be expressed through the roots of certain equations.A similar expression can be written down in the case N oc, but there is only one root to be found.More general results, such as ergodicity and heavy traffic limit theorems, were considered by Borovkov [2, 3].
Our aim is to generalize the results concerning the "classical" model to the case of the queueing systems M/MR/1/N and GI/MR/1/N with renovation. 1 1We preserve the Kendall notation for a queueing system with renovation, equipping the second letter with superscript R.
These queueing systems are of significant interest for our applications.The first publications to study such models were papers by Kreinin [11] and by Towsley and  Tripathi [13].The motivation of these models is given below.
Kreinin [11] deals with a model of a so-called "prefetch instruction buffer".The prefetch buffer provides a pipeline mechanism in the most recent processors and microprocessors for systems with a single instruction flow.This architectural element improves the performance when the linear part of a program is running by allowing the simultaneous processing of independent parts of different instructions.If the recent instruction is "branch" (or "go to';), then the processor loses the content of the prefetch instruction buffer and renovates it.
Another example of a renovation mechanism is considered by Towsley and Tri-  pathi in [13].They studied the probabilistic models of buddy protocols for fault-toler- ant systems.This model [13] leads to priority queues with renovation (in [13] the term "flushing" was used) which does not depend on the service process.
This paper is organized as follows.In Section 2 we discuss the ergodicity condi- tions for general queueing systems with renovation.In Section 3 we consider a sys- tem with renovation under the simplest assumptions on the interarrival and service times.Both cases, N < cx3 and N , are considered.Section 4 contains the results for the case of generally distributed interarrival times and Markovian service.Final remarks and some directions for further studies are discussed in Section 5.

Ergodicity Conditions
Consider a queueing system with renovation and general interarrival and service time distributions.Assume that the customers arrive from the outside at moments tl, t2,... and form a recurrent flow with interarrival times In-tn+l-tn, n-0,1, Thus, the sequence {In} forms a family of independent, identically distributed, random variables (r.v.'s) with a common distribution function F(x).Assume < EI-/ xdF(x) < 0 0 Let S n be the service time of the nth customer.Assume that {Sn} are independent and identically distributed r.v.'s with ES 1 < Denote by (t, the queue length at time t, (t >_ 0).Assume that the initial queue length is finite with probability 1.
Proposition 1: If 0 < p < 1, then for each initial state o of the queue satisfying the condition Pr{ o < oo} 1, the queueing system with renovation has a unique stationary distribution 7rj -tlimPr{t j}, j O, 1,..., which does not depend on the initial state of the queue length process.
Proof: Denote by A {inf 0 < s < ts 0} the event that the queue once became empty during [0, t].The process-t-has a renovating event {t 0} (see norovkov [3]).For all finite values of 0, the probability of the complementary event tends to zero inf t s > O} t At { o < < Pt-Pr{t}-*O, as t---,c.Indeed, let N(t, t + T) be the number of customers served without renova- tion of the queue.Then for each e > 0 and for each integer M, there exists T suffi- ciently large, such that for every t > 0, Pr{N(t, t + T) > M} < e.Thus, the probabili- ty of the event .A tends to 1 as t--cx.The existence of the renovating event pro- vides the ergodicity of the queueing system.
This results shows that the renovation probability is a critical parameter for the stability domain.Indeed, if q > 0, EI 1 < cx, and ES 1 < cx, the system is stable.In the case q 0, the system is stable if and only if the inequality EI 1 > ESI is satis- fied.
3. System with Poisson Input Flow Consider a queueing system with renovation.We assume that the system has only one queue with FIFO discipline and one input flow of customers.The input flow is assumed to be Poisson.Denote by ,k the input flow intensity.Service times are mutually independent and exponentially distributed random variables with parameter #.
After a customer is serviced, all others can leave the queue with probability q.Only a customer who receives service leaves the system with probability p 1-q.
Below we consider two classes of queues with renovation, Poisson input flow, and ex- ponential service time distribution.
3.1 M/Mlt/N Queueing System Denote by N the size of the waiting room.system at moment t.Then t -< N. Denote Let t be the number of customers in the rj(t)-Pr{t-j}, j-O,I,...,N.
The stationary distribution of the queue length is given by the following assertion.
Proposition 2: If 0 < p < 1 and # > O, then for each initial state of the queue satisfying the condition wj-limPr{ t-j} (j-O,1 N).

t--oo
The generating function of this distribution satisfies the equation r(z where Q(z)-.z 2-( + #)z + p#, and the probabilities o and r N are given by the formulas Z *N zN, with z* and z, being the roots of the equation Q(z)-O.
Proof: The process t is a Markov chain.It is easy to verify that for 0 < p < 1 and # > 0 this Markov chain is ergodic.The stationary distribution {Trj}jc=0 of t satisfies the equations and the normalization condition Then, from (1) after standard calculations we obtain (1) The equation Q(z)-0 has two roots, z, and z*.Both roots are real and positive. 2The generating function r(z) is a polynomial.The stationary probabilities r 0 and r N must satisfy the equations Itpr0(1 z*)-ATrN z*N + 1(1 z*) Itqz*.
(3) From (2) and (3), we obtain the relations for r 0 and r N. Proposition 2 is proved. [:]  Corollary 1: If the renovation probability q-O, then the stationary distribution of the queue length is given by 7r k 7ropk 1, k 1,..., N,

1-p 1 pN'
where p A/it.If the renovation probability q 1, then the stationary distribution of the queue length is given by # ( A )k, k-0, l...N-l, -A+it A+# Proof: The nt part of the corollary is well known (see e.g., [1,8]).The second It is not difficult to see that for p > 0 these roots satisfy the inequalities z* > 1, 0<z,<l.
part follows directly from the equations (1) for the equilibrium distribution of the queue length process.
3.2 MX/MR/I/oo Queueing System Let us study now the case N cx.We consider a more general situation of bulk input.Assume that the customers arrive in groups of random size, and interarrival times are mutually independent, exponentially distributed r.v.'s with parameter A. This arrival flow is usually called a compound Poisson process.Denote Pk--Pr{1--k}, k--l,2,..., where is the size of an arrival group of customers.We denote by p(z) the generating function of this distribution.Thus, ()-z , k=0 where zl _< 1.The average size of the arriving groups is assumed to be finite.
Then the function p(z) is regular in the unit disc D-{z: z < 1} and the average size of the arrival group is equal to El-p'(1).
Proposition 3" If 0 < p < 1, El < cxz, and # > O, then for each initial state, the queueing system has a unique stationary queue length distribution.The generating function of this distribution is given by the formula #p(1 z)Tr o q#z (z) Q,(z) where Q(z) zp(z)-( + ,)z + p,, qZ, r p(1 z ,)' and z, is a unique root of the equation Qp(z)-0, that satisfies the inequality 0<z,<l.
Proof: Denote by qm (m-1,2,...) the number of customers in the system at the arrival epoch 7-m of the ruth customer.Using the standard technique it is not difficult to verify that {qm} is a positive supermartingale.On the other hand, the queue length t at the moment t, 7" m < t 7 m _1_ 1 satisfies the inequality 0 <_ t <qm for m >_ 0. The last inequality implies that the process t has a stationary distribution 7r n limtPr{ n} that satisfies the equations / + # Tr k " E 7r k n P n + #P Tr k + 1' k l, 2,..., (4) n--1 N 0 + m j. (5)

General Arrival Process
In this section we consider a queueing system with renovation, general distribution of the interarrival times In, and exponential service time distribution.To describe the behavior of the system we will use an intensity function ,(x), and parameter # of the service time distribution.The intensity function satisfies the relations F(x)-exp A(t)dt .(x)>_ O, 0 where 1-F(x)-F(x).The behavior of the queue length process is described by 3 semi-Markov process.
4.1 GI/MR/1/N Queue with Renovation and Lost Customers Consider a queueing system with finite waiting room and renovation.Let the process t-(kt, xt, t > 0), k 7/+, x N +, where k is the number of customers in the system (k _< N), and x is the elapsed time since the last arrival at moment t.The process (kt, xt) is a Markov process.
It is known (see, e.g., [6]) that there exist the densities of the probability distribution pn(t,x OPn(t'x) cox 3Examples of the general theory of semi-Markov processes and their application in queueing theory may be found in the books by Gnedenko and Kovalenko [5] and by Jaiswal [6]; see also the book by inlar [4].
and their stationary limits pn(x) lim OPn(t'x) t--,oo Ox n-O,1,...,N, z >_ O. (7) The densities Pn(X) do not depend on the initial state of the process t" If the stationary densities Pn(X) are known, the stationary probabilities can be easily found in the form: (8) 0 There are two different ways to control the system when the queue buffer is saturated (k N).The first is to lose the arriving customer, while the second is to block the input flow.We consider the case of the system with lost customers.To formulate the desired assertion we need the following notations: and Let F n e-"xxnF(x)dx, 7n et'XxndF(x), n O, 1,..., The stationary densities of the queue length process in the system with lost customers are given by N-k 1,...,N, i=0 where the coefficients Ci, k satisfy the relations where the coefficients C m satisfy the relations Relations (8)-( 13) define the ergodic distribution of system GI/MR/1/N.
It can easily be seen that boundary condition ( 23) is met.
Using expression (24) we can rewrite boundary condition ( 22) as (25 Therefore, a should satisfy the equation -XdF(x), which is equivalent to (19).Standard arguments concerning the convexity of the Laplace transform lead to the conclusion that equation ( 19) has only one root in the interval (0,1).
To find c, we express the stationary probabilities r k through this unknown constant and then, use the normalization condition j 0r/ 1. Namely, for k _> 1 we have rk / pk(x)dx cat j e-X (x)dx

Final Remarks
The purpose of this paper is to present the results for queueing system with renova- tion in Markov case, and in the case of the general recurrent arrival flow.Using the same technique, one can study the case of Poisson arrival and generally distributed service times.The calculations should be performed in this case using the method of embedded Markov chains.In Section 4.1 we studied the GI/MR/1/N queueing system with renovation and lost customers.In the same rnanner one can study the queueing system with blocking input when the queue is saturated.
More complicated problems arise when the interarrival and service times have general distributions.Almost nothing is known for such systems.The problem of calculating stationary characteristics has remained open.In this case, the approach based on heavy traffic limit theorems should be fruitful.The approximation for the waiting time for the customers in heavy traffic has the form of a diffusion process in the line modified to include jumps to origin at random moments.Such processes appeared in the paper by Kella and Whitt [7], devoted to the diffusion approxima- tions for queues with server vacations.
Another direction for generalizing the model discussed in this paper is a multi- phase queueing system with Poisson arrival flows and exponentially distributed ser- vice times.This queue leads to non-homogeneous random walks that can be studied in several important cases.This problem will be considered in future publications. .
formula and the normalization condition, we obtain the constant c.Thus, the relation (18) is proved.I-!