Table of Contents
Journal of Applied Mathematics and Stochastic Analysis
VolumeΒ 2009, Article IDΒ 308025, 14 pages
http://dx.doi.org/10.1155/2009/308025
Research Article

Interloss Time in 𝑀/𝑀/1/1 Loss System

Occupational Medicine Department (DML), National Institute for Occupational Safety and Prevention (ISPESL), Via Alessandria 220/E, 00198 Rome, Italy

Received 29 March 2009; Accepted 24 May 2009

Academic Editor: HoΒ Lee

Copyright Β© 2009 Pierpaolo Ferrante. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the interloss times in the 𝑀/𝑀/1/1 Erlang Loss System. Here we present the explicit form of the probability density function of the time spent between two consecutive losses in the 𝑀/𝑀/1/1 model. This density function solves a Cauchy problem for the second-order differential equations, which was used to evaluate the corresponding laplace transform. Finally the connection between the Erlang's loss rate and the evaluated probability density function is showed.

1. Introduction

In this paper, we treat the random variable 𝑇(𝑖) representing the time spent between the 𝑖th and the (π‘–βˆ’1)th lost unit or 𝑖th interloss time, in the 𝑀/𝑀/1/1 loss system.

The 𝑀/𝑀/1/1 model is characterized by the Markov property of entering and exiting processes, by one service channel and by the system capacity to accommodate one customer at a time (for an overview see Medhi [1, page. 77]).

Our work has been inspired by the location problem of emergency vehicles (ambulances). Each vehicle can be regarded as an 𝑀/𝑀/1/1 (or 𝑀/πΈπ‘˜/1/1) system, because its clients cannot wait in the queue.

In Emergency Medical Systems (EMSs), the nearest ambulance to the accident place is called β€œdistrict unit”, and it assures the best performance to the system. If an emergency call arrives at EMS while its β€œdistrict unit” is busy, the nearest ambulance among those available is dispatched (see Larson [2] or Larson [3]). The length of interloss times affects the performance of the system and provides an informative support on efficiency of EMS.

For the exponential 𝑀/𝑀/1/1 loss model it was conducted, in Ferrante [4], a detailed description of the process of losses {𝐿(𝑑)}𝑑>0,(1.1) where 𝐿(𝑑) represent the random number of losses in the time interval [0,𝑑). Let π‘ƒπ‘š(π‘˜)(𝑑) be the conditional probability to lose π‘š clients in [0,𝑑) with π‘˜ customers in the system at time 𝑑=0: π‘ƒπ‘š(π‘˜)(𝑑)=𝑃{𝐿(𝑑)=π‘šβˆ£π‘˜},π‘˜=0,1,(1.2) the main results found in Ferrante [4] are the explicit values of the conditional probabilities of no losses in [0,𝑑): 𝑃0(1)(𝑑)=π‘’βˆ’(𝑑/2)(2πœ†+πœ‡)ξƒ¬π‘‘βˆšcoshπœ‡(4πœ†+πœ‡)2+πœ‡βˆšπ‘‘βˆšπœ‡(4πœ†+πœ‡)sinhπœ‡(4πœ†+πœ‡)2ξƒ­,𝑃0(0)(𝑑)=π‘’βˆ’(𝑑/2)(2πœ†+πœ‡)ξƒ¬π‘‘βˆšcoshπœ‡(4πœ†+πœ‡)2+2πœ†+πœ‡βˆšπ‘‘βˆšπœ‡(4πœ†+πœ‡)sinhπœ‡(4πœ†+πœ‡)2ξƒ­,(1.3) and the iterative procedure to determine the distribution of the total number of losses in [0,𝑑).All is obtained by solving the inhomogeneous differential equations π‘ƒπ‘š(π‘˜)𝑑(𝑑)=βˆ’(2πœ†+πœ‡)π‘ƒπ‘‘π‘‘π‘š(π‘˜)(𝑑)βˆ’πœ†2π‘ƒπ‘š(π‘˜)𝑑(𝑑)+πœ†π‘ƒπ‘‘π‘‘(π‘˜)π‘šβˆ’1(𝑑)+πœ†2𝑃(π‘˜)π‘šβˆ’1,(1.4) with 𝑃(π‘˜)βˆ’1(𝑑)=0 and the initial conditions depending on the π‘˜ (=0,1) customers in the system at 𝑑=0. Furthermore, the generating probability functions πΊπ‘˜(𝑠,𝑑) of π‘ƒπ‘š(π‘˜)(𝑑) were evaluated and their explicit values are the following: 𝐺1(𝑠,𝑑)=π‘’βˆ’(𝑑/2)(2πœ†+πœ‡βˆ’πœ†π‘ )Γ—ξƒ¬π‘‘βˆšcosh(πœ†π‘ βˆ’πœ‡)2+4πœ†πœ‡2+πœ‡+πœ†π‘ βˆš(πœ†π‘ βˆ’πœ‡)2π‘‘βˆš+4πœ†πœ‡sinh(πœ†π‘ βˆ’πœ‡)2+4πœ†πœ‡2ξƒ­,𝐺0(𝑠,𝑑)=π‘’βˆ’(𝑑/2)(2πœ†+πœ‡βˆ’πœ†π‘ )Γ—ξƒ¬π‘‘βˆšcosh(πœ†π‘ βˆ’πœ‡)2+4πœ†πœ‡2+2πœ†+πœ‡βˆ’πœ†π‘ βˆš(πœ†π‘ βˆ’πœ‡)2π‘‘βˆš+4πœ†πœ‡sinh(πœ†π‘ βˆ’πœ‡)2+4πœ†πœ‡2ξƒ­.(1.5)

The aim of this work is to identify the type of the process of interloss times 𝑇(𝑖)ξ€Ύπ‘–βˆˆπ‘+(1.6) for the 𝑀/𝑀/1/1 loss model and to find the differential equation which governs it, in order to determine the probability density functions 𝑓𝑇(𝑖)𝑑(𝑑)=𝑃𝑇𝑑𝑑(𝑖)ξ€Ύ,<𝑑(1.7) with 𝑖=1,2,… and the related properties.

Our results show unexpected connections among very different branches of probability such as random motion on hyperbolic space and queueing systems. In effect, the probabilities appearing below have a structure quite similar to the Hyperbolic distances of moving particles envisaged in Cammarota and Orsingher [5].

In Section 2, we establish that {𝑇(𝑖)}π‘–βˆˆπ‘+ is a renewal process for the 𝑀(πœ†)/𝑀(πœ‡)/1/1 loss system and that the density functions (1.7) solve the second-order linear homogeneous differential equations 𝑑2𝑑𝑑2𝑓𝑇(𝑖)𝑑(𝑑)=βˆ’(2πœ†+πœ‡)𝑓𝑑𝑑𝑇(𝑖)(𝑑)βˆ’πœ†2𝑓𝑇(𝑖)(𝑑).(1.8)

Let 𝜈(𝑑) be the number of customers in the system at the moment 𝑑, and let 𝑑𝑙𝑖 be the moment of the 𝑖th loss with 𝑑𝑙0=0. The initial conditions for (1.8) depend on 𝜈(π‘‘π‘™π‘–βˆ’1), and the renewal process {𝑇(𝑖)}π‘–βˆˆπ‘+ has the following property:

𝑓𝑇(1)(𝑑)=𝑓𝑇(𝑖)(𝑑),ifπ‘“πœˆ(0)=1,𝑇(1)(𝑑)≠𝑓𝑇(𝑖)(𝑑),if𝜈(0)=0(1.9) for 𝑖>1.

In Section 2, we also present the derivation of (1.8) and its solution conditionally by 𝜈(0).

Let 𝑓𝑇(1)(𝑑;𝑖) be the conditional density function of the 1th interloss time with 𝑖 (= 0,1) customers in the system at time 𝑑=0: 𝑓𝑇(1)𝑑(𝑑;𝑖)=𝑃𝑇𝑑𝑑(1)ξ€Ύ.<π‘‘βˆ£πœˆ(0)=𝑖(1.10)

For the 𝑀(πœ†)/𝑀(πœ‡)/1/1 model, the explicit values obtained for (1.10) are the following: 𝑓𝑇(1)(𝑑;1)=πœ†π‘’βˆ’(𝑑/2)(2πœ†+πœ‡)ξƒ¬π‘‘βˆšcoshπœ‡(4πœ†+πœ‡)2βˆ’πœ‡βˆšπ‘‘βˆšπœ‡(4πœ†+πœ‡)sinhπœ‡(4πœ†+πœ‡)2ξƒ­,𝑓𝑇(1)(𝑑;0)=2πœ†2π‘’βˆ’(𝑑/2)(2πœ†+πœ‡)βˆšπœ‡π‘‘βˆš(4πœ†+πœ‡)sinhπœ‡(4πœ†+πœ‡)2.(1.11)

In Section 3, we compute the laplace transforms of (1.10): πΉβˆ—π‘‡(1)ξ€œ(𝑠;𝑖)=∞0π‘’βˆ’π‘ π‘‘π‘“π‘‡(1)(𝑑;𝑖)𝑑𝑑(1.12) for 𝑖=0,1, using (1.8).

The explicit values obtained for (1.12) are the following:

πΉβˆ—π‘‡(1)(𝑠;1)=πœ†(πœ†+𝑠)(πœ†+𝑠)2,𝐹+π‘ πœ‡βˆ—π‘‡(1)πœ†(𝑠;0)=2(πœ†+𝑠)2.+π‘ πœ‡(1.13)

Finally, let Θ1(𝑖) be the conditional means of the 1th interloss time Θ1(𝑖)𝑇=𝐸(1)ξ€»,∣𝜈(0)=𝑖(1.14) with 𝑖=0,1; it has been checked that their values are Θ1(1)=1π‘Ÿ,Θ1(0)=1π‘Ÿ+1πœ†,(1.15) where r is the Erlang loss rate, and πœ†βˆ’1 is the interarrival mean time.

2. First Interloss Time

In the 𝑀(πœ†)/𝑀(πœ‡)/1/1 model, let 𝜈(𝑑) be the number of customers in the system at the moment 𝑑, let 𝜏(π‘˜) be the π‘˜th interarrival time, let π‘‘π‘˜ be the moment when the π‘˜th client enters the system, let π‘†π‘˜ be the service time of the π‘˜th served customer, let 𝑇(𝑖) be the 𝑖th interloss time. Furthermore, let 𝑙𝑖(𝑑) be the arrival order of 𝑖th loss happened in 𝑑, starting from the (π‘–βˆ’1)th loss, and let 𝑑𝑙𝑖 be the moment when the 𝑖th loss happen, with 𝑑𝑙0=0 and 𝑑𝑙𝑖(𝑑)=𝑑.

If we consider that the system is busy at time 𝑑=0, the event β€œThe 1th interloss time is 𝑑” is represented by Figure 1.

308025.fig.001
Figure 1: The 1th interloss time is 𝑑 with 𝑃{𝜈(0)=1}=1.

The random variable 𝑇(1) can be expressed as follows: 𝑇(1)=𝑙1(𝑑)ξ“π‘˜=1𝜏(π‘˜),(2.1) where 𝑙1(𝑑) represents the arrival order of the 1th loss happened in 𝑑𝑙1(𝑑)=𝑑, starting from zero.

Now, let 𝑃𝑛(1)(𝑑) be the conditional probability that the arrival order of 1th loss happened in 𝑑 with 𝜈(0)=1 is equal to π‘›βˆΆπ‘ƒπ‘›(1)𝑙(𝑑)=𝑃1ξ€Ύ,(𝑑)=π‘›βˆ£πœˆ(0)=1(2.2) and it can be expressed as follows 𝑃𝑛(1)𝑆(𝑑)=𝑃0<𝜏(1),…,π‘†π‘›βˆ’2<𝜏(π‘›βˆ’1),π‘†π‘›βˆ’1>π‘‘βˆ’π‘‘π‘›βˆ’1ξ€Ύ(2.3) and can be computed conditionally by the (π‘›βˆ’1) moments when the served customers have arrived at the system 𝑃𝑛(1)(𝑑)=(π‘›βˆ’1)!π‘‘π‘›βˆ’1ξ€œπ‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’2π‘‘π‘‘π‘›βˆ’1π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›βˆ’1)π‘›βˆ’1𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»=(π‘›βˆ’1)!π‘‘π‘›βˆ’1𝐹(0)π‘›βˆ’1,1(𝑑),(2.4) where 𝐹(0)π‘›βˆ’1,1ξ€œ(𝑑)=𝑑0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’2π‘‘π‘‘π‘›βˆ’1π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›βˆ’1)π‘›βˆ’1𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€».(2.5)

Lemma 2.1. The functions 𝐹(𝑠)𝑛,1ξ€œ(𝑑)=𝑑𝑠𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’1π‘‘π‘‘π‘›π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›)𝑛𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»(2.6) with 𝑑0=𝑠 do not depend on t but on the time interval [𝑠,𝑑): 𝐹(𝑠)𝑛,1(𝑑)=𝐹(0)𝑛,1(π‘‘βˆ’π‘ ).(2.7)

Proof. We proceed by showing that (2.7) is true for 𝑛=1: 𝐹(𝑠)1,1(ξ€œπ‘‘)=𝑑𝑠𝑑𝑑1π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘1)ξ€Ί1βˆ’π‘’βˆ’πœ‡(𝑑1βˆ’π‘ )ξ€»=1βˆ’π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )πœ‡βˆ’(π‘‘βˆ’π‘ )π‘’βˆ’πœ‡(π‘‘βˆ’π‘ )=𝐹(0)1,1(π‘‘βˆ’π‘ ).(2.8)
Then, we suppose that it is true for π‘›βˆ’1, and we obtain that
𝐹(𝑠)𝑛,1(ξ€œπ‘‘)=𝑑𝑠𝑑𝑑1ξ€Ί1βˆ’π‘’βˆ’πœ‡(𝑑1βˆ’π‘ )𝐹(𝑑1)π‘›βˆ’1,1(=ξ€œπ‘‘)𝑑𝑠𝑑𝑑1𝐺1(𝑠)𝑑1𝐹(0)π‘›βˆ’1,1ξ€·π‘‘βˆ’π‘‘1ξ€Έ,(2.9) where 𝐺1(𝑠)𝑑1ξ€Έ=ξ€œπ‘‘1𝑠𝑑π‘₯πœ‡π‘’βˆ’πœ‡(𝑑1βˆ’π‘₯).(2.10)
Finally, by the Markov property of the exponential distribution, the (2.7) appears
𝐹(𝑠)𝑛,1(ξ€œπ‘‘)=0π‘‘βˆ’π‘ π‘‘π‘‘1𝐺1(0)𝑑1𝐹(0)π‘›βˆ’1,1ξ€·π‘‘βˆ’π‘ βˆ’π‘‘1ξ€Έ.(2.11)

The conditional density function 𝑓𝑇(1)𝑑(𝑑;1)=𝑃𝑇𝑑𝑑(1)ξ€Ύ<π‘‘βˆ£πœˆ(0)=1(2.12) can be evaluated as mean of convolution of 𝑙1(𝑑) exponential probability density functions, and thus we have that 𝑓𝑇(1)(𝑑;1)=βˆžξ“π‘›=1π‘“βˆ‘π‘›π‘˜=1𝜏(π‘˜)(𝑑)𝑃𝑛(1)(𝑑)=π‘’βˆžβˆ’πœ†π‘‘ξ“π‘›=0πœ†π‘›+1𝐹(0)𝑛,1(𝑑).(2.13) At first, we state the following result concerning the evaluation of the integrals 𝐹(0)𝑛,1(𝑑),𝑛β‰₯1. Lemma 2.2. The functions 𝐹(0)𝑛,1ξ€œ(𝑑)=𝑑0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’1π‘‘π‘‘π‘›π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›)𝑛𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»(2.14) satisfy the difference-differential equations 𝑑2𝑑𝑑2𝐹(0)𝑛,1𝑑(𝑑)=βˆ’πœ‡πΉπ‘‘π‘‘(0)𝑛,1(𝑑)+πœ‡πΉ(0)π‘›βˆ’1,1(𝑑),(2.15) where 𝑑0=0, 𝑑>0,𝑛β‰₯1.Proof. We first note that 𝑑𝐹𝑑𝑑(0)𝑛,1𝑑(𝑑)=ξ€œπ‘‘π‘‘π‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’1π‘‘π‘‘π‘›π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›)𝑛𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»=ξ€œπ‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’2π‘‘π‘‘π‘›βˆ’1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›βˆ’1)ξ€»π‘›βˆ’1𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»ξ€œβˆ’πœ‡π‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’1π‘‘π‘‘π‘›π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›)𝑛𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»,(2.16) and therefore 𝑑2𝑑𝑑2𝐹(0)𝑛,1ξ€œ(𝑑)=πœ‡π‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’2π‘‘π‘‘π‘›βˆ’1π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›βˆ’1)π‘›βˆ’1𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»ξ€œβˆ’πœ‡π‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’2π‘‘π‘‘π‘›βˆ’1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›βˆ’1)ξ€»π‘›βˆ’1𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»+πœ‡2ξ€œπ‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’1π‘‘π‘‘π‘›π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›)𝑛𝑖=1ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)𝑑=βˆ’πœ‡πΉπ‘‘π‘‘(0)𝑛,1(𝑑)+πœ‡πΉ(0)π‘›βˆ’1,1(𝑑).(2.17)

In view of Lemma 2.2 we can prove also the following.

Theorem 2.3. The function 𝑓𝑇(1)(𝑑;1) satisfies the second-order linear homogeneous differential equation 𝑑2𝑑𝑑2𝑓𝑇(1)𝑑(𝑑;1)=βˆ’(2πœ†+πœ‡)𝑓𝑑𝑑𝑇(1)(𝑑;1)βˆ’πœ†2𝑓𝑇(1)(𝑑;1),(2.18) with the initial conditions 𝑓𝑇(1)𝑑(0;1)=πœ†,𝑓𝑑𝑑𝑇(1)(𝑑;1)|𝑑=0=βˆ’πœ†(πœ†+πœ‡).(2.19) The explicit value of 𝑓𝑇(1)(𝑑;1) is 𝑓𝑇(1)(𝑑;1)=πœ†π‘’βˆ’(𝑑/2)(2πœ†+πœ‡)ξƒ¬βˆšπœ‡(4πœ†+πœ‡)βˆ’πœ‡2βˆšπ‘’πœ‡(4πœ†+πœ‡)√(𝑑/2)πœ‡(4πœ†+πœ‡)+βˆšπœ‡(4πœ†+πœ‡)+πœ‡2βˆšπ‘’πœ‡(4πœ†+πœ‡)βˆšβˆ’(𝑑/2)πœ‡(4πœ†+πœ‡)ξƒ­.(2.20)

Proof. From (2.13), it follows that 𝑑𝑓𝑑𝑑𝑇(1)(𝑑;1)=βˆ’πœ†π‘“π‘‡(1)(𝑑;1)+π‘’βˆžβˆ’πœ†π‘‘ξ“π‘›=0πœ†π‘›+1𝑑𝐹𝑑𝑑(0)𝑛,1(𝑑),(2.21) and thus, in view of (2.17) and by letting 𝐹(0)βˆ’1,1(𝑑)=0 , we have that 𝑑2𝑑𝑑2𝑓𝑇(1)(𝑑𝑑;1)=βˆ’πœ†π‘“π‘‘π‘‘π‘‡(1)(𝑑;1)βˆ’πœ†π‘’βˆžβˆ’πœ†π‘‘ξ“π‘›=0πœ†π‘›+1𝑑𝐹𝑑𝑑(0)𝑛,1(𝑑)+π‘’βˆžβˆ’πœ†π‘‘ξ“π‘›=0πœ†π‘›+1𝑑2𝑑𝑑2𝐹(0)𝑛,1(𝑑𝑑)=βˆ’2πœ†π‘“π‘‘π‘‘π‘‡(1)(𝑑;1)βˆ’πœ†2𝑓𝑇(1)(𝑑;1)+πœ†πœ‡π‘“π‘‡(1)𝑑(𝑑;1)βˆ’πœ‡π‘“π‘‘π‘‘π‘‡(1)(𝑑;1)+πœ†π‘“π‘‡(1)𝑑(𝑑;1)=βˆ’(2πœ†+πœ‡)𝑓𝑑𝑑𝑇(1)(𝑑;1)βˆ’πœ†2𝑓𝑇(1)(𝑑;1).(2.22) While the first condition is straightforward to verify, the second one needs some explanations: if we write 𝑑𝑓𝑑𝑑𝑇(1)||(𝑑;1)𝑑=0=limΔ𝑑→0𝑓𝑇(1)(Δ𝑑;1)βˆ’π‘“π‘‡(1)(0;1),Δ𝑑(2.23) and observe that 𝑓𝑇(1)(Δ𝑑;1)=π‘’βˆ’πœ†Ξ”π‘‘πœ†πΉ(0)0,1(Δ𝑑)=πœ†π‘’βˆ’(πœ†+πœ‡)Δ𝑑[]=πœ†1βˆ’(πœ†+πœ‡)Δ𝑑+π‘œ(Δ𝑑),(2.24) by substituting (2.24) in (2.23) the second condition emerges.
The general solution to (2.22) has the form
π‘’βˆ’(𝑑/2)(2πœ†+πœ‡)ξ‚ƒπ΄π‘’βˆš(𝑑/2)πœ‡(4πœ†+πœ‡)+π΅π‘’βˆšβˆ’(𝑑/2)πœ‡(4πœ†+πœ‡)ξ‚„.(2.25) By imposing the initial conditions (2.19) to (2.18) we obtain (2.20).

Remark 2.4. By (2.7) derive that the functions 𝑓𝑇(𝑖)(𝑑;1) do not depend on t, but on the time interval [π‘‘π‘™π‘–βˆ’1,𝑑), in fact if π‘‘π‘™π‘–βˆ’1=𝑠, we have that 𝑓𝑇(𝑖)(𝑑;1,𝑠)=π‘’βˆžβˆ’πœ†(π‘‘βˆ’π‘ )𝑛=0πœ†π‘›+1𝐹(𝑠)𝑛,1(𝑑)=𝑓𝑇(𝑖)(π‘‘βˆ’π‘ ;1),(2.26) for 𝑖=1,2,…
Furthermore, by the Markov properties of the 𝑀/𝑀/1/1 system, the random variables 𝑇(1),𝑇(2),… are independent, and {𝑇(𝑖)}π‘–βˆˆπ‘+ is a renewal process with
𝑓𝑇(1)(𝑑)=𝑓𝑇(𝑖)(𝑑)ifπ‘“πœˆ(0)=1,𝑇(1)(𝑑)≠𝑓𝑇(𝑖)(𝑑)if𝜈(0)=0.(2.27)

Remark 2.5. If πœ†β†’0 we get 𝑓𝑇(1)(𝑑;1)=0, because the 1th interloss time is greater than 𝑑, βˆ¨π‘‘>0, when nobody enters.
If πœ‡β†’0, we have that 𝑓𝑇(1)(𝑑;1)=πœ†π‘’βˆ’πœ†π‘‘ because without exits and with the system busy at 𝑑=0, the 1th interloss time has the same distribution of the interarrival time.

Remark 2.6. The probability density function 𝑓𝑇(1)(𝑑;1) can be expressed by the following hyperbolic functions: 𝑓𝑇(1)(𝑑;1)=πœ†π‘’(βˆ’π‘‘/2)(2πœ†+πœ‡)ξƒ¬π‘‘βˆšcoshπœ‡(4πœ†+πœ‡)2βˆ’πœ‡βˆšπ‘‘βˆšπœ‡(4πœ†+πœ‡)sinhπœ‡(4πœ†+πœ‡)2ξƒ­.(2.28)

Now, if we assume that the system is free at the starting point, the event β€œThe 1th interloss time is 𝑑” is represented by Figure 2.

308025.fig.002
Figure 2: The 1th interloss time is 𝑑 with 𝑃{𝜈(0)=0}=1.

Let 𝑃𝑛(0)(𝑑) be the conditional probability that the nth entered customer is the 1th lost at time t, when 𝜈(0)=0, 𝑃𝑛(0)𝑙(𝑑)=𝑃1𝑆(𝑑)=π‘›βˆ£πœˆ(0)=0=𝑃1<𝜏(2),…,π‘†π‘›βˆ’2<𝜏(π‘›βˆ’1),π‘†π‘›βˆ’1>π‘‘βˆ’π‘‘π‘›βˆ’1ξ€Ύ,(2.29) it can be computed conditionally by the (π‘›βˆ’1) moments when the served customers have arrived at the system 𝑃𝑛(0)(𝑑)=(π‘›βˆ’1)!π‘‘π‘›βˆ’1ξ€œπ‘‘0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’2π‘‘π‘‘π‘›βˆ’1π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›βˆ’1)π‘›βˆ’1𝑖=2ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»=(π‘›βˆ’1)!π‘‘π‘›βˆ’1𝐹(0)π‘›βˆ’1,0(𝑑),(2.30) where 𝐹(0)π‘›βˆ’1,0ξ€œ(𝑑)=𝑑0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’2π‘‘π‘‘π‘›βˆ’1π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›βˆ’1)π‘›βˆ’1𝑖=2ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€».(2.31) The function 𝑓𝑇(1)(𝑑;0) can be computed as mean of convolution of 𝑙1(𝑑) exponential probability density functions. So we have that 𝑓𝑇(1)(𝑑;0)=βˆžξ“π‘›=2π‘“βˆ‘π‘›π‘˜=1𝜏(π‘˜)(𝑑)𝑃𝑛(0)(𝑑)=π‘’βˆžβˆ’πœ†π‘‘ξ“π‘›=2πœ†π‘›πΉ(0)π‘›βˆ’1,0(𝑑).(2.32)

Lemma 2.7. The functions 𝐹(0)𝑛,0ξ€œ(𝑑)=𝑑0𝑑𝑑1β‹―ξ€œπ‘‘π‘‘π‘›βˆ’1π‘‘π‘‘π‘›π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›)𝑛𝑖=2ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»(2.33) satisfy the difference-differential equations 𝑑2𝑑𝑑2𝐹(0)𝑛,0𝑑(𝑑)=βˆ’πœ‡πΉπ‘‘π‘‘(0)𝑛,0(𝑑)+πœ‡πΉ(0)π‘›βˆ’1,0(𝑑),(2.34) where 𝑑>0,𝑛β‰₯1,𝐹(0)0,0(𝑑)=0.Proof. See proof of Lemma 2.2.

In view of Lemma 2.7 we can prove also the following.

Theorem 2.8. The function 𝑓𝑇(1)(𝑑;0) satisfies the second-order linear homogeneous differential equation 𝑑2𝑑𝑑2𝑓𝑇(1)𝑑(𝑑;0)=βˆ’(2πœ†+πœ‡)𝑓𝑑𝑑𝑇(1)(𝑑;0)βˆ’πœ†2𝑓𝑇(1)(𝑑;0),(2.35) with the initial conditions 𝑓𝑇(1)𝑑(0;0)=0,𝑓𝑑𝑑𝑇(1)||(𝑑;0)𝑑=0=πœ†2.(2.36) The explicit value of 𝑓𝑇(1)(𝑑;0) is 𝑓𝑇(1)πœ†(𝑑;0)=2βˆšπ‘’πœ‡(4πœ†+πœ‡)βˆ’(𝑑/2)(2πœ†+πœ‡)ξ‚ƒπ‘’βˆš(𝑑/2)πœ‡(4πœ†+πœ‡)βˆ’π‘’βˆšβˆ’(𝑑/2)πœ‡(4πœ†+πœ‡)ξ‚„.(2.37)

Proof. By substituting in (2.21) and (2.22) 𝐹(0)𝑛,1(𝑑) with 𝐹(0)𝑛,0(𝑑), (2.35) emerges. While the first condition is straightforward to verify, the second one needs some explanations: if we write 𝑑𝑓𝑑𝑑𝑇(1)||(𝑑;0)𝑑=0=limΔ𝑑→0𝑓𝑇(1)(Δ𝑑;0)βˆ’π‘“π‘‡(1)(0;0),Δ𝑑(2.38) and observe that 𝑓𝑇(1)(Δ𝑑;0)=π‘’βˆ’πœ†Ξ”π‘‘πœ†2π‘’βˆ’πœ‡Ξ”π‘‘ξ€œ0Ξ”π‘‘π‘’πœ‡π‘‘1𝑑𝑑1=πœ†2Δ𝑑+π‘œ(Δ𝑑),(2.39) by substituting (2.39) in (2.38), the second condition emerges.
By imposing the initial conditions (2.36) to (2.35), we obtain (2.37).

Remark 2.9. If πœ†β†’0, we get 𝑓𝑇(1)(𝑑;0)=0 because the 1th interloss time is greater than 𝑑, βˆ¨π‘‘>0, when nobody enters.
If πœ‡β†’0, we have that 𝑓𝑇(1)(𝑑;0)=πœ†2π‘‘π‘’βˆ’πœ†π‘‘ because without exits and with the system free at 𝑑=0, the 1th interloss time is equal to the sum of 1th and 2th interarrival times.

Remark 2.10. The function 𝑓𝑇(1)(𝑑;0) can be expressed by the following hyperbolic function: 𝑓𝑇(1)(𝑑;0)=2πœ†2π‘’βˆ’(𝑑/2)(2πœ†+πœ‡)βˆšπ‘‘βˆšπœ‡(4πœ†+πœ‡)sinhπœ‡(4πœ†+πœ‡)2.(2.40)

Remark 2.11. The function 𝑓𝑇(1)(𝑑;0) can also be computed as the convolution of the exponential density with rate πœ† and 𝑓𝑇(1)(𝑑;1): 𝑓𝑇(1)(ξ€œπ‘‘;0)=𝑑0πœ†π‘’βˆ’πœ†π‘‘1𝑓𝑇(1)ξ€·π‘‘βˆ’π‘‘1ξ€Έ;1𝑑𝑑1.(2.41)

In fact, if we observe that 𝐹(0)𝑛,0ξ€œ(𝑑)=𝑑0𝑑𝑑1ξ€œπ‘‘π‘‘1𝑑𝑑2β‹―ξ€œπ‘‘π‘‘π‘›βˆ’1π‘‘π‘‘π‘›π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘π‘›)𝑛𝑖=2ξ€Ί1βˆ’π‘’βˆ’πœ‡(π‘‘π‘–βˆ’π‘‘π‘–βˆ’1)ξ€»=ξ€œπ‘‘0𝑑𝑑1𝐹(𝑑1)π‘›βˆ’1,1(𝑑),(2.42) by (2.7), (2.32), and (2.42) we obtain that 𝑓𝑇(1)(𝑑;0)=π‘’βˆžβˆ’πœ†π‘‘ξ“π‘›=2πœ†π‘›ξ€œπ‘‘0𝑑𝑑1𝐹(0)π‘›βˆ’2,1ξ€·π‘‘βˆ’π‘‘1ξ€Έ=π‘“πœ(1)βˆ—π‘“π‘‡(1)(𝑑;1).(2.43)

3. Interloss Mean Time

In this section we compute the laplace transforms of the density functions (2.20) and (2.37), and we evaluate the averages of the 1th interloss time, conditionally by 𝜈(0).

Theorem 3.1. The laplace transform of 𝑓𝑇(1)(𝑑;1)πΉβˆ—π‘‡(1)ξ€œ(𝑠;1)=∞0π‘’βˆ’π‘ π‘‘π‘“π‘‡(1)(𝑑;1)𝑑𝑑(3.1) satisfies the equation 𝑠2πΉβˆ—π‘‡(1)(𝑠;1)βˆ’π‘ πœ†+πœ†(πœ†+πœ‡)=βˆ’(2πœ†+πœ‡)π‘ πΉβˆ—π‘‡(1)(𝑠;1)+πœ†(2πœ†+πœ‡)βˆ’πœ†2πΉβˆ—π‘‡(1)(𝑠;1),(3.2) and its explicit value is πΉβˆ—π‘‡(1)(𝑠;1)=πœ†(πœ†+𝑠)(πœ†+𝑠)2.+π‘ πœ‡(3.3)

Proof. By (2.18), (2.19), and the property ξ€œβˆž0π‘‘π‘‘π‘’βˆ’π‘ π‘‘π‘‘π‘“π‘‘π‘‘π‘‡(1)(𝑑;1)=π‘ πΉβˆ—π‘‡(1)(𝑠;1)βˆ’π‘“π‘‡(1)(0;1),(3.4) the results (3.2) and (3.3) emerge.

Remark 3.2. The 1th conditional interloss mean time Θ1(1)𝑇=𝐸(1)ξ€»βˆ£πœˆ(0)=1(3.5) can be found by evaluating the derivative of (3.3) with respect to 𝑠 in 𝑠=0, as follows: βˆ’π‘‘πΉπ‘‘π‘ βˆ—π‘‡(1)||(𝑠;1)𝑠=0=πœ†+πœ‡πœ†2.(3.6) In the 𝑀/𝑀/1/1 model, the interloss mean time (3.5) is equal to the inverse of the Erlang loss rate.

Theorem 3.3. The laplace transform of 𝑓𝑇(1)(𝑑;0)πΉβˆ—π‘‡(1)ξ€œ(𝑠;0)=∞0π‘’βˆ’π‘ π‘‘π‘“π‘‡(1)(𝑑;0)𝑑𝑑(3.7) satisfies the equation 𝑠2πΉβˆ—π‘‡(1)(𝑠;0)βˆ’πœ†2=βˆ’(2πœ†+πœ‡)π‘ πΉβˆ—π‘‡(1)(𝑠;0)βˆ’πœ†2πΉβˆ—π‘‡(1)(𝑠;0),(3.8) and its explicit value is πΉβˆ—π‘‡(1)πœ†(𝑠;0)=2(πœ†+𝑠)2.+π‘ πœ‡(3.9)

Proof. By (2.35), (2.36), and (3.4), the results (3.8) and (3.9) emerge.

Remark 3.4. The conditional interloss mean time Θ1(0)𝑇=𝐸(1)ξ€»βˆ£πœˆ(0)=0(3.10) can be found by evaluating the derivative of (3.9) with respect to 𝑠 in 𝑠=0, as follows: βˆ’π‘‘πΉπ‘‘π‘ βˆ—π‘‡(1)||(𝑠;0)𝑠=0=Θ1(1)+1πœ†.(3.11)

Remark 3.5. By (2.41), the result (3.9) can be obtained using (3.3), as follows: πΉβˆ—π‘‡(1)πœ†(𝑠;0)=πΉπœ†+π‘ βˆ—π‘‡(1)(𝑠;1).(3.12)

Now, let 𝑇𝑛 be the time spent between 𝑑=0 and the 𝑛th loss, the conditional density functions 𝑓𝑇𝑛𝑑(𝑑;𝑖)=𝑃𝑇𝑑𝑑𝑛,<𝑑;𝑖(3.13) with 𝑖=0,1, can be evaluated by convolutions as follows: if 𝑖=0, we have that 𝑓𝑇𝑛(𝑑;0)=π‘“πœ(1)βˆ—π‘“π‘‡(1)(𝑑;1)βˆ—β‹―βˆ—π‘“π‘‡(𝑛)(𝑑;1),(3.14) while if 𝑖=1, we have that 𝑓𝑇𝑛(𝑑;1)=𝑓𝑇(1)(𝑑;1)βˆ—β‹―βˆ—π‘“π‘‡(𝑛)(𝑑;1).(3.15) By the independence between interloss times, the laplace transforms of (3.14) and (3.15) can be expressed as power of (3.3) as follows:

(i)if 𝑖=0, πΉβˆ—π‘‡π‘›(𝑠;0)=πΉβˆ—πœ(0)+βˆ‘π‘›π‘˜=1𝑇(π‘˜)πœ†(𝑠;1)=ξ‚Έπœ†+π‘ πœ†(πœ†+𝑠)(πœ†+𝑠)2ξ‚Ή+πœ‡π‘ π‘›,(3.16)(ii) if 𝑖=1, πΉβˆ—π‘‡π‘›(𝑠;1)=πΉβˆ—βˆ‘π‘›π‘˜=1𝑇(π‘˜)ξ‚Έ(𝑠;1)=πœ†(πœ†+𝑠)(πœ†+𝑠)2ξ‚Ή+πœ‡π‘ π‘›.(3.17)

Remark 3.6. The conditional averages Ξ˜π‘›(𝑖)𝑇=𝐸𝑛,∣𝜈(0)=𝑖(3.18) with 𝑖=0,1, can be permuted by evaluating the derivatives of (3.16) and (3.17) with respect to 𝑠 in 𝑠=0, thus obtaining(i)if 𝑖=0, we have that βˆ’π‘‘πΉπ‘‘π‘ βˆ—π‘‡π‘›||(𝑠;0)𝑠=0=π‘›Ξ˜1(1)+1πœ†,(3.19)(ii)if 𝑖=1, we have that βˆ’π‘‘πΉπ‘‘π‘ βˆ—π‘‡π‘›||(𝑠;1)𝑠=0=π‘›Ξ˜1(1).(3.20)

Acknowledgments

The author wants to acknowledge his PhD tutor Professor Enzo Orsingher for the theoretical support and Professor Francesca Guerriero and Dr. Patrizia Scano for their contributions.

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