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Mathematics 2018, 6(9), 159; https://doi.org/10.3390/math6090159
Fractional Queues with Catastrophes and Their Transient Behaviour
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, 80126 Napoli, Italy
School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
Author to whom correspondence should be addressed.
Received: 27 July 2018 / Accepted: 31 August 2018 / Published: 6 September 2018
Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al. in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al. in 2003, we define and study a fractional M/M/1 queue with catastrophes. In particular, we focus our attention on the transient behaviour, in which the time-change plays a key role. We first specify the conditions for the global uniqueness of solutions of the corresponding linear fractional differential problem. Then, we provide an alternative expression for the transient distribution of the fractional M/M/1 model, the state probabilities for the fractional queue with catastrophes, the distributions of the busy period for fractional queues without and with catastrophes and, finally, the distribution of the time of the first occurrence of a catastrophe.
Keywords:fractional differential-difference equations; fractional queues; fractional birth-death processes; busy period
Stochastic models for queueing systems have a wide range of applications in computer systems, sales points, telephone or telematic systems and also in several areas of science including biology, medicine and many others. The well known M/M/1 queueing model [1,2,3,4,5] constitutes the theoretical basis for building many other refined models for service systems.
Due to the Markov nature of its inter-arrival times of the customers and of its service times, the model can be mathematically treated in a simple manner, and, for this reason, it is widely used in many modeling contexts. Nevertheless, in the past few decades, the advent of fractional operators, such as fractional derivatives and integrals (see, for instance,  and  and references therein), has made it clear that different time scales, themselves random, that preserve memory (therefore not Markovian), allow the construction of more realistic stochastic models.
The introduction of the fractional Caputo derivative into the system of differential-difference equations for an M/M/1-type queue was done in , where, for a fractional M/M/1 queue, the state probabilities were determined. In this kind of queue model, the inter-arrival times and service times are characterized by Mittag–Leffler distributions ; in this case, the model does not have the property of memory loss that is typical of the exponential distributed times of the classical M/M/1 model. Indeed, a time-changed birth-death process [10,11], by means of an inverse stable subordinator , solves the corresponding fractional system of differential-difference equations and fractional Poisson processes  characterize the inter-arrival and service times.
The fractional M/M/1 model in  is an interesting and powerful model, not only because it is a generalization of the classical one, where the fractional order is set to 1, but also because its range of applications is extremely wide. Its importance can be further augmented by including in the model the occurrence of catastrophes, as it was considered in  for the classical M/M/1.
The catastrophe is a particular event that occurs in a random time leading to the instantaneous emptying of the system, or to a momentary inactivation of the system, as, for example, the action of a virus program that can make a computer system inactive ; other applications of models with catastrophes can be found in population dynamics and reliability contexts (see  and references therein).
Motivated by the mathematical need to enrich the fractional M/M/1 model of  with the inclusion of catastrophes, we study in this paper the above model; specifically, we determine the transient distribution, the distribution of the busy period (including that of the fractional M/M/1 queue of ) and the probability distribution of the time of the first occurrence of the catastrophe.
For these purposes, we need to guarantee the global uniqueness of the solution of the considered linear fractional Cauchy problem on Banach spaces. After recalling the definitions and known results in Section 2, we address the problem of uniqueness in Section 3. In Section 2, we also provide the transient distribution of the fractional M/M/1 model in an alternative form to that given in . In Section 4, the distribution of the busy period for the fractional M/M/1 queue (without catastrophes) is obtained. Here, the time-changed birth-death process plays a key role to derive the results. In Section 5, we define the fractional queue with catastrophes; we are able to obtain the distribution of the transient state probabilities by following a strategy similar to that in . We also found the distribution of the busy period and of the time of the first occurrence of the catastrophe starting from the empty system. Some special operators and functions used in this paper are specified in the Appendix A and Appendix B.
2. Definition of a Fractional Process Related to M/M/1 Queues
The classical M/M/1 queue process can be described as continuous time Markov chain whose state space is the set and the state probabilitiessatisfy the following differential-difference equations:where is the Kroeneker delta symbol, and are the entrance and service rates, respectively.
Let be the Lévy -stable subordinator with Laplace exponent given by:
Consider the inverse -stable subordinator
For , the fractional M/M/1 queue process is defined by a non-Markovian time change independent of , i.e.,
This process was defined in  and it is non-Markovian with non-stationary and non-independent increments. For , by definition, . Then, for a fixed , the state probabilitiesof the number of customers in the system at time t in the fractional M/M/1 queue are characterized by arrivals and services determined by fractional Poisson processes of order  with parameters and . They are solutions of the following system of differential-difference equationswhere is the Caputo fractional derivative (see Appendix A).
Using Equation (5) and representation (3), the state probabilities are obtained in :as well as its Laplace transform
In Equation (6), the functions are generalized Mittag–Leffler functions (see Appendix B). Note that and .
Alternatively, let be the density of ; then it is known (see, i.e., ) thatand (see, i.e., , Proposition 4.1)
Using (7) and an analytical expression for given in , we can write down an alternative expression for (6) aswhere , , and is the -th derivative of the function evaluated at .
3. Linear Fractional Cauchy Problems on Banach Spaces
In order to describe the transient probabilities for our queues, we will need some uniqueness results for solutions of linear fractional Cauchy problems defined on Banach spaces. To do that, let us recall the following Theorem (Theorem from ):
Let be a Banach space and for some . Consider the ball . Let and and consider the following Cauchy problem:where is the Caputo derivative operator (see Appendix A).
- There exists a constant such that
- There exists a constant such that ;
- There exists a constant such that
- There exist constants and such that
The previous theorem can be easily adapted to the case in which and the starting point of the derivative is . Since we are interested in linear (eventually non-homogeneous) equations, let us show how the previous theorem can be adapted in such a case.
Consider the system (10) and suppose where is a linear and continuous operator and . Then, there exists a and such that the system admits a unique solution .
Observe that, if , then
Let us choose T such that the conditions of Theorem 1 are verified. To do that, consider . Fix and define for some . Define thenand observe that
Thus, one can fix and . Moreover, since for fixed the function is decreasing and , then one can easily find a such that . Since we are under the hypotheses of Theorem 1, then we have shown the local existence and uniqueness of a solution . ☐
However, using such corollary, we can only afford local uniqueness. Global uniqueness of the solution of the Cauchy problem (10) can be obtained with the additional hypothesis that such solution is uniformly bounded:
Suppose we are under the hypotheses of Corollary 1. If there exists a solution and a constant such that for any we have , then such solution is unique.
Observe that and then fix . Define
Fix and observe that, by using Corollary 1, there exists a unique solution in . Since is a solution of such problem, we have that is unique. Suppose we have defined such that is the unique solution of system (10) in . Consider the problem
Define then and observe that, since , by using Corollary 1, there exists a unique solution in .
4. The Fractional M/M/1 Queue
Consider the Hilbert space with the norm and let be the space of the -Hölder continuous functions from to . One can rewrite the system (5) in as follows:where andis an infinite tridiagonal matrix with . Let us show the following:
The linear operator is continuous and .
To show that is continuous, let us use Schur’s test (Theorem in ). Observe thatso that, in general,
Moreover,so that, in general,
By Schur’s test, we have that is a bounded operator on and☐
Thus, by Corollary 1, we obtain local existence and uniqueness of the solution of system (5). Global uniqueness can be obtained a posteriori, since the solutions of such system are known.
Let us also observe that the distributions of the inter-arrival times are Mittag–Leffler distributions. To do that, consider the system, for fixedwhich are the state probabilities of a queue with null death rate, fixed birth rate, starting with n customers and with an absorbent state . Under such assumptions, is the probability that a customer arrives before t. Moreover, the normalizing condition becomes
One can solve the first equation (see Appendix A) to obtainwhere is the one-parameter Mittag–Leffler function (see Appendix B), and then, by using the normalizing condition, we have
In a similar way, let us show that the distributions of the service times are Mittag–Leffler distributions. To show that, consider the system, for fixed ,which are the state probabilities of a queue with null birth rate, fixed death rate, starting with customers with an absorbent state n. Under such assumption, is the probability that a customer is served before t. Moreover, the normalizing condition becomes
One can solve the second equation to obtainand then, by using the normalizing condition, we have
Moreover, since we know that and , by the continuous inclusion (see ), is uniformly bounded in and then, by Corollary 2, it is the (global) unique solution of system (5).
Distribution of the Busy Period
We want to determine the probability distribution of the busy period of a fractional M/M/1 queue. To do this, we will follow the lines of the proof given in  and .
Let be the random variable describing the duration of the busy period of a fractional M/M/1 queue and consider . Then,where
Let us first define a queue such that and behaves like except for the state 0 being an absorbent state. Thus, state probability functions are solution of the following system
First, we want to show that, if we consider the inverse of a -stable subordinator that is independent from , then . To do that, consider the probability generating function of defined as
From system (15), we know that solves the following fractional Cauchy problem:which, for , becomes
Taking the Laplace transform in Equation (17) and using Equation (A1), we havewhere and are Laplace transforms of and .
We know that if and only ifand then if and only if, by Equation (16),
Taking the Laplace transform in Equations (20) and (21) for and by using (see, i.e., Equation (10) in )we know we have to show thatand
Since Equation (17) admits a unique solution, then we only need to show that the right-hand sides of Equations (23) and (24) solve Equation (19), that is to say that we have to verify
To do that, consider the right-hand side of the previous equation and, recalling that is solution of Equation (18):and then, by integrating by parts, we have Equation (25).
Now remark that . Thus, we want to determine . To do that, let us recall, from [1,4] thatfrom which, explicitly writing , we have
Posing , we haveand then
Since , we haveand then, using Equation (26), we have
Finally, using formula (A2), we have☐
As we obtain, by usingthat and then .
5. The Fractional M/M/1 Queue with Catastrophes
Let us consider a classical M/M/1 queue with FIFO discipline and subject to catastrophes whose effect is to instantaneously empty the queue  and let be the number of customers in the system at time t with state probabilities
Then, the function satisfy the following differential-difference equations:where is the Kroeneker delta symbol, , are the entrance and service rates, respectively, and is the rate of the catastrophes when the system is not empty.
For , we define the fractional M/M/1 queue process with catastrophes aswhere is an inverse -stable subordinator that is independent of (see Section 2).
We will show that the state probabilitiessatisfy the following differential-difference fractional equations:where is the Caputo fractional derivative (see Appendix A).
In the classical case, catastrophes occur according to a Poisson process with rate if the system is not empty. In our case, consider for a fixed ,which describes the state probabilities of an initially non empty system with null birth and death rate but positive catastrophe rate. In such case, is the probability a catastrophe occurs before time t. Moreover, the normalization property becomes
In such case, we can solve the second equation to obtain
Using the normalization property, we finally obtainand then the distributions of the inter-occurrence of the catastrophes are Mittag–Leffler distributions. We can conclude that, in the fractional case, catastrophes occur according to a fractional Poisson process ([10,11,13]) with rate if the system is not empty. Since the operators are Caputo fractional derivatives, we expect the stationary behaviour of the queue to be the same as the classic one. Denoting with the number of customers in the system at the steady state of a classical M/M/1 with catastrophes and defining the state probabilitieswe can use the results obtained in  to observe thatwhere is the solution ofsuch that . Let us call the other solution of Equation (32) and observe that . Some properties coming from such equations that will be useful hereafter areandwith .
5.1. Alternative Representation of the Fractional M/M/1 Queue with Catastrophes
We want to obtain an alternative representation of the fractional M/M/1 queue with catastrophes in a way which is similar to Lemma in . To do that, we firstly need to assure that system (29) admits a unique uniformly bounded solution. To do that, let us write system (29) in the formwhere , , andis an infinite tridiagonal matrix with . We need to show the following:
The linear operator is continuous and .
To obtain an estimate of the norm of , let us use Schur’s test. Observe thatso that, in general,
Moreover,so that, in general,
By Schur’s test, we have that is a bounded operator on and
Observe that, if , the operator is the same of system (12). Let us also observe that by Corollary 1 there locally exists a unique solution. Moreover, if we show that a solution is uniformly bounded, such solution is unique. Now, we are ready to adapt Lemma 2.1 of  to the fractional case.
Let be the number of customers in a fractional M/M/1 queue with arrival rate and service rate such that and consider N a random variable independent from whose state probabilities are defined in Equation (31). DefineThen, the state probabilities of are the unique solutions of (29).
Moreover, , where is the equality in distribution, and then are the unique solutions of (29).
Define and . Since and N are independent, thenwhich, by using the definitions of and , becomes
We want to show that . Since, by definition, are non-negative and , they are uniformly bounded in . Thus, we only need to show that solves system (35).
The initial conditions are easily verified, so we only need to verify the differential relations. Observe thatand then, applying the Caputo derivative operator, we obtain
Since is a solution of system (5) with rates and , we have
Observe also thatso we have
After some calculations, we obtain
Let us remark thatso we have
Rewrite now Equation (38) in the formand then apply a Caputo derivative operator to obtain
Since is a solution of system (5) with birth rate and death rate , then we have
Remark that, by using Equation (39),
Then, recalling that by definition and and doing some calculations, we have
We have shown that the state probabilities of are the unique solutions of system (29). Now, we need to show that . To do this, consider a classical M/M/1 queue with arrival rate and service rate , N a random variable independent from and with probability masses and finally the inverse of a -stable subordinator which is independent from N and . Define also . By Lemma of , we know that , so . However, by definition, we know that , thus finally☐
5.2. State Probabilities for the Fractional M/M/1 with Catastrophes
Since we have defined , where is the inverse of a -stable subordinator, which is independent from , we can use such definition and Theorem 3 with the results obtained in  to study the state probabilities of . In particular, we refer to the formulawhere
By using such formula, we can show the following:
For any and , we havewhere are defined in (41).
From , we haveand then, by using formula (40),
Taking the Laplace transform and using Equation (22), we obtainand then, integrating
Finally, by using Equation (A2), we obtain☐
From formula (42), we can easily see that so, as we expected, the steady-state probabilities are the same as the classical ones. For such reason, we can say that the fractional behaviour is influential only in the transient state of the queue.
As , by usingwe obtain that .
If and , then and . For such reason, , and . Then, we have that of Equation (42) has the form of Equation (6).
If and then and . In such case, , and . For such case, we havewhich is not recognizable as a previously obtained formula. This is due to the fact that the formula(which is the one that is obtained from (42) as and , as done in ) has no known equivalent in the fractional case. It is also interesting to observe that in  another representation of the Laplace transform of is given in formula , which is not easily invertible, but has been obtained by using (43) instead of Sharma’s representation of ()
5.3. Distribution of the Busy Period
Let denote the duration of the busy period and be its probability distribution function. Let us observe that, if we pose , then the queue empties within t if and only if a catastrophe occurs within t or otherwise the queue empties without catastrophes within t. Let us remark that, if there is no occurrence of catastrophes, the queue behaves as a fractional M/M/1. Let us define as the duration of a busy period for a fractional M/M/1 queue without catastrophes, the time of first occurrence of a catastrophe for a non empty queue and and their probability distribution functions. Thus, we have
By using formula (44), we can finally show:
Let be the duration of the busy period of a fractional M/M/1 queue with catastrophes and . Then,where is given in (14).
5.4. Distribution of the Time of the First Occurrence of a Catastrophe
We already know that if the queue starts from a non-empty state, then the occurrence of the catastrophes is a Mittag–Leffler distribution. However, we are interested in such distribution as the queue starts being empty. To do that, we will need some auxiliary discrete processes.
Let be the time of first occurrence of a catastrophe as and let . Then,where
Following the lines of , let us consider the process with state space such that and posing as its state probability, we have
Let us remark that such process represents our queue until a catastrophe occurs: in such case, instead of emptying the queue, the state of the process becomes , which is an absorbent state. With such interpretation, we can easily observe that .
In order to determine , we will first show that where is the inverse of a -stable subordinator which is independent from . To do that, let us consider instead of . Let us remark that . Let be the probability generating function of . Multiplying the third sequence of equations in (47) with and then, summing all these equations, we have
Now observe thatmoreover,finally,
We have obtained that the probability generating function of solves the Cauchy problemthat, for , becomes
Let , and be the Laplace transforms of , and and let us take the Laplace transform in Equation (53) to obtain
Now, let us remark that if and only if for all :that is to say if and only if
Taking Laplace transform and using Equation (22), we obtain
Finally, multiplying with , we have
Now we know that if and only if Equation (58) is verified. For this reason, we only need to show such equation. To do that, remarking that , consider the right-hand side of Equation (58) and observe that
Thus, by using Equation (54), we haveconcluding the proof of our first claim.
From Theorem of , we know thatand, since we know that , we can use (59) in (56) with to obtain:
Our work focused on the transient behaviour of a fractional M/M/1 queue with catastrophes, deriving formulas for the state probabilities, the distribution of the busy period and the distribution of the time of the first occurrence of a catastrophe. This is a non-Markov generalization of the classical M/M/1 queue with catastrophes, obtained through a time-change. The introduction of fractional dynamics in the equations that master the behaviour of the queue led to a sort of transformation of the time scale. Fractional derivatives are global operators, so the state probabilities preserve memory of their past, eventually slowing down the entire dynamics. Indeed, we can see how Mittag–Leffler functions take place where in the classical case we expected to see exponentials. However, such fractional dynamic seems to affect only the transient behaviour, since we have shown in Remark 2 that the limit behaviour is the same.
The main difficulty that is linked with fractional queues (or in general time-changed queues) is the fact that one has to deal with non-local derivative operators, such as the Caputo derivative, losing Markov property. However, fractional dynamics and fractional processes are gaining attention, due to their wide range of applicability, from physics to finance, from computer science to biology. Moreover, time-changed processes have formed a thriving field of application in mathematical finance. Future works will focus on an extension of such results to and queues, or even to a generalization of fractional M/M/1 queue to a time-changed M/M/1 queue by using the inverse of any subordinator.
The three authors have participated equally in the development of this work. The paper was also written and reviewed cooperatively.
This research was partially funded by GNCS and MANM. Nikolai Leonenko was supported in parts by the project MTM2015–71839–P(cofounded with Feder funds), of the DGI, MINECO, and the Australian Research Council’s Discovery Projects funding scheme (project number DP160101366).
We want to thank V. V. Ahn of QUT, Brisbane, Australia for his support.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. Fractional Integrals and Derivatives
Let us recall the definition of fractional integral . Given a function , its fractional integral of order in for is given by:
The Riemann–Liouville fractional derivative operator is defined as:while the Caputo fractional derivative operator is defined as:whenever . Obviously, such operators are linear. It is interesting to remark that
Note that, for a function and , the Caputo fractional derivative is defined as:whereand for its Laplace transform, denoting by the Laplace transform of x,
Moreover, for , a well-posed fractional Cauchy problem with Riemann–Liouville derivatives is given in the formin which the initial condition is given in terms of fractional integrals, while if we use Caputo derivatives we have:in which the initial condition is related only with the initial value of the function. For such reason, we will prefer adopting Caputo derivatives as fractional derivatives in this paper.
Finally, let us remark that the definition of fractional integral and derivative can be also considered for functions where B is a Banach space and all the involved integrals are Bochner integrals ().
Appendix B. Some Special Functions
We recall the definitions of some special functions we use in such text.
The Gamma function is defined as:
In particular, we have and, for , .
The modified Bessel function () of the first kind can be defined by its power series expansion as:
One-parameter Mittag–Leffler functions () are defined by their power series expansion as:
Two-parameters Mittag–Leffler functions are also defined by their power series expansion as:
Remark that .
Generalized Mittag–Leffler functions are defined by their power series expansion as:where is the Pochhammer symbol
An alternative way to define the Generalized Mittag–Leffler function is:
Remark also that . Functions with similar series expansions are also involved in the study of the asymptotic behaviour of some integrals, which arise from a Feynman path integral approach to some financial problems (see, i.e.,  Section 4).
Recall also the following Laplace transform formula 
Finally let us remark, for that the Cauchy problemadmits as unique solution (, p. 295).
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