Interloss Time in 𝑀/𝑀/1/1 Loss System

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 (𝑖−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/𝐸_𝑘/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 “district unit”, and it assures the best performance to the system. If an emergency call arrives at EMS while its “district 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(𝑡)}_𝑡>0,(1.1) where 𝐿(𝑡) represent the random number of losses in the time interval [0,𝑡). Let 𝑃^𝑚(𝑘)(𝑡) be the conditional probability to lose 𝑚 clients in [0,𝑡) with 𝑘 customers in the system at time 𝑡=0: 𝑃^𝑚(𝑘)(𝑡)=𝑃{𝐿(𝑡)=𝑚∣𝑘},𝑘=0,1,(1.2) the main results found in Ferrante [4] are the explicit values of the conditional probabilities of no losses in [0,𝑡): 𝑃⁰⁽¹⁾(𝑡)=𝑒^{−(𝑡/2)(2𝜆+𝜇)}cosh𝑡√𝜇(4𝜆+𝜇)2+√𝜇(4𝜇𝜆+𝜇)sinh𝑡√𝜇(4𝜆+𝜇)𝑃₀₍₀₎2,(𝑡)=𝑒^{−(𝑡/2)(2𝜆+𝜇)}cosh𝑡√𝜇(4𝜆+𝜇)2+√𝜇2𝜆(4+𝜆+𝜇𝜇)sinh𝑡√𝜇(4𝜆+𝜇)2,(1.3) and the iterative procedure to determine the distribution of the total number of losses in [0,𝑡).All is obtained by solving the inhomogeneous differential equations 𝑃^𝑚(𝑘)(𝑡)=−(2𝜆+𝜇)𝑑𝑑𝑡𝑃^𝑚(𝑘)(𝑡)−𝜆²𝑃^𝑚(𝑘)(𝑡)+𝜆𝑑𝑑𝑡𝑃^{(𝑘)𝑚−1}(𝑡)+𝜆²𝑃^{(𝑘)𝑚−1},(1.4) with 𝑃^(𝑘)−1(𝑡)=0 and the initial conditions depending on the 𝑘 (=0,1) customers in the system at 𝑡=0. Furthermore, the generating probability functions 𝐺_𝑘(𝑠,𝑡) of 𝑃^𝑚(𝑘)(𝑡) were evaluated and their explicit values are the following: 𝐺₁(𝑠,𝑡)=𝑒^{−(𝑡/2)(2𝜆+𝜇−𝜆𝑠)}×cosh𝑡√(𝜆𝑠−𝜇)²+4𝜆𝜇2+√(𝜆𝑠𝜇−+𝜇)𝜆²𝑠+4𝜆𝜇sinh𝑡√(𝜆𝑠−𝜇)²+4𝜆𝜇𝐺₀2,(𝑠,𝑡)=𝑒^{−(𝑡/2)(2𝜆+𝜇−𝜆𝑠)}×cosh𝑡√(𝜆𝑠−𝜇)²+4𝜆𝜇2+√(𝜆2𝜆𝑠−+𝜇𝜇)−²+𝜆𝑠4𝜆𝜇sinh𝑡√(𝜆𝑠−𝜇)²+4𝜆𝜇2.(1.5)

The aim of this work is to identify the type of the process of interloss times 𝑇_(𝑖)_𝑖∈𝑍⁺(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 𝑓_𝑇(𝑖)(𝑡)=𝑑𝑑𝑡𝑃𝑇^(𝑖)<𝑡,(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)}_𝑖∈𝑍⁺ is a renewal process for the 𝑀(𝜆)/𝑀(𝜇)/1/1 loss system and that the density functions (1.7) solve the second-order linear homogeneous differential equations 𝑑²𝑑𝑡²𝑓_𝑇(𝑖)(𝑡)=−(2𝜆+𝜇)𝑑𝑑𝑡𝑓^𝑇(𝑖)(𝑡)−𝜆²𝑓_𝑇(𝑖)(𝑡).(1.8)

Let 𝜈(𝑡) be the number of customers in the system at the moment 𝑡, and let 𝑡_𝑙𝑖 be the moment of the 𝑖th loss with 𝑡_𝑙0=0. The initial conditions for (1.8) depend on 𝜈(𝑡_𝑙𝑖−1), and the renewal process {𝑇_(𝑖)}_𝑖∈𝑍⁺ has the following property:

𝑓𝑓_𝑇(1)(𝑡)=𝑓_𝑇(𝑖)(𝑡),if𝜈(0)=1,_𝑇(1)(𝑡)≠𝑓_𝑇(𝑖)(𝑡),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 𝑓_𝑇(1)(𝑡;𝑖) be the conditional density function of the 1th interloss time with 𝑖 (= 0,1) customers in the system at time 𝑡=0: 𝑓_𝑇(1)(𝑡;𝑖)=𝑑𝑑𝑡𝑃𝑇⁽¹⁾<𝑡∣𝜈(0)=𝑖.(1.10)

For the 𝑀(𝜆)/𝑀(𝜇)/1/1 model, the explicit values obtained for (1.10) are the following: 𝑓_𝑇(1)(𝑡;1)=𝜆𝑒^{−(𝑡/2)(2𝜆+𝜇)}cosh𝑡√𝜇(4𝜆+𝜇)2−√𝜇(4𝜇𝜆+𝜇)sinh𝑡√𝜇(4𝜆+𝜇)𝑓_𝑇(1)2,(𝑡;0)=2𝜆²𝑒^{−(𝑡/2)(2𝜆+𝜇)}√𝜇(4𝜆+𝜇)sinh𝑡√𝜇(4𝜆+𝜇)2.(1.11)

In Section 3, we compute the laplace transforms of (1.10): 𝐹^∗𝑇₍₁₎(𝑠;𝑖)=^∞0𝑒^−𝑠𝑡𝑓_𝑇(1)(𝑡;𝑖)𝑑𝑡(1.12) for 𝑖=0,1, using (1.8).

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

𝐹^∗𝑇₍₁₎(𝑠;1)=𝜆(𝜆+𝑠)𝐹(𝜆+𝑠)²+𝑠𝜇_∗𝑇(1),(𝑠;0)=𝜆²(𝜆+𝑠)²+𝑠𝜇.(1.13)

Finally, let Θ^1(𝑖) be the conditional means of the 1th interloss time Θ^1(𝑖)=𝐸𝑇₍₁₎∣𝜈(0)=𝑖,(1.14) with 𝑖=0,1; it has been checked that their values are Θ¹⁽¹⁾=1𝑟,Θ¹⁽⁰⁾=1𝑟+𝜆1,(1.15) where r is the Erlang loss rate, and 𝜆⁻¹ is the interarrival mean time.

2. First Interloss Time

In the 𝑀(𝜆)/𝑀(𝜇)/1/1 model, let 𝜈(𝑡) be the number of customers in the system at the moment 𝑡, let 𝜏_(𝑘) be the 𝑘th interarrival time, let 𝑡_𝑘 be the moment when the 𝑘th client enters the system, let 𝑆_𝑘 be the service time of the 𝑘th served customer, let 𝑇_(𝑖) be the 𝑖th interloss time. Furthermore, let 𝑙_𝑖(𝑡) be the arrival order of 𝑖th loss happened in 𝑡, starting from the (𝑖−1)th loss, and let 𝑡_𝑙𝑖 be the moment when the 𝑖th loss happen, with 𝑡_𝑙0=0 and 𝑡_{𝑙𝑖(𝑡)}=𝑡.

If we consider that the system is busy at time 𝑡=0, the event “The 1th interloss time is 𝑡” is represented by Figure 1.

Figure 1 

The 1th interloss time is $t$ with $P\{\nu (0)=1\}=1.$

The random variable 𝑇₍₁₎ can be expressed as follows: 𝑇₍₁₎=^𝑙1(𝑡)_𝑘=1𝜏_(𝑘),(2.1) where 𝑙₁(𝑡) represents the arrival order of the 1th loss happened in 𝑡_𝑙1(𝑡)=𝑡, starting from zero.

Now, let 𝑃^𝑛(1)(𝑡) be the conditional probability that the arrival order of 1th loss happened in 𝑡 with 𝜈(0)=1 is equal to 𝑛∶𝑃^𝑛(1)(𝑡)=𝑃𝑙₁(𝑡)=𝑛∣𝜈(0)=1,(2.2) and it can be expressed as follows 𝑃^𝑛(1)(𝑡)=𝑃𝑆₀<𝜏₍₁₎,…,𝑆_𝑛−2<𝜏_(𝑛−1),𝑆_𝑛−1>𝑡−𝑡_𝑛−1(2.3) and can be computed conditionally by the (𝑛−1) moments when the served customers have arrived at the system 𝑃^𝑛(1)(𝑡)=(𝑛−1)!𝑡^𝑛−1^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−2𝑑𝑡_𝑛−1𝑒^{−𝜇(𝑡−𝑡𝑛−1)𝑛−1}=(𝑛−1)!_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}𝑡^𝑛−1𝐹^{(0)𝑛−1,1}(𝑡),(2.4) where 𝐹^{(0)𝑛−1,1}(𝑡)=^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−2𝑑𝑡_𝑛−1𝑒^{−𝜇(𝑡−𝑡𝑛−1)𝑛−1}_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}.(2.5)

Lemma 2.1. The functions 𝐹^(𝑠)𝑛,1(𝑡)=^𝑡𝑠𝑑𝑡₁⋯^𝑡𝑡_𝑛−1𝑑𝑡_𝑛𝑒^{−𝜇(𝑡−𝑡𝑛)𝑛}_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}(2.6) with 𝑡₀=𝑠 do not depend on t but on the time interval [𝑠,𝑡): 𝐹^(𝑠)𝑛,1(𝑡)=𝐹^(0)𝑛,1(𝑡−𝑠).(2.7)

Proof. We proceed by showing that (2.7) is true for 𝑛=1: 𝐹^(𝑠)1,1(𝑡)==1^𝑡𝑠−𝑑𝑡𝑒₁𝑒^{−𝜇(𝑡−𝑡1)}_{−𝜇(𝑡−𝑠)}1−𝑒^{−𝜇(𝑡1−𝑠)}𝜇−(𝑡−𝑠)𝑒^{−𝜇(𝑡−𝑠)}=𝐹^(0)1,1(𝑡−𝑠).(2.8)
Then, we suppose that it is true for 𝑛−1, and we obtain that
𝐹^(𝑠)𝑛,1(𝑡)==^𝑡𝑠𝑑𝑡₁1−𝑒^{−𝜇(𝑡1−𝑠)}𝐹^{(𝑡1)𝑛−1,1}(𝑡)_𝑡𝑠𝑑𝑡₁𝐺^1(𝑠)𝑡₁𝐹^{(0)𝑛−1,1}𝑡−𝑡₁,(2.9) where 𝐺^1(𝑠)𝑡₁=^𝑡1_𝑠𝑑𝑥𝜇𝑒^{−𝜇(𝑡1−𝑥)}.(2.10)
Finally, by the Markov property of the exponential distribution, the (2.7) appears
𝐹^(𝑠)𝑛,1(𝑡)=^0𝑡−𝑠𝑑𝑡₁𝐺¹⁽⁰⁾𝑡₁𝐹^{(0)𝑛−1,1}𝑡−𝑠−𝑡₁.(2.11)

The conditional density function 𝑓_𝑇(1)(𝑡;1)=𝑑𝑑𝑡𝑃𝑇⁽¹⁾<𝑡∣𝜈(0)=1(2.12) can be evaluated as mean of convolution of 𝑙₁(𝑡) exponential probability density functions, and thus we have that 𝑓_𝑇(1)(𝑡;1)=^∞_𝑛=1𝑓_∑^𝑛𝑘=1_𝜏(𝑘)(𝑡)𝑃^𝑛(1)(𝑡)=𝑒^{∞−𝜆𝑡}_𝑛=0𝜆^𝑛+1𝐹^(0)𝑛,1(𝑡).(2.13) At first, we state the following result concerning the evaluation of the integrals 𝐹^(0)𝑛,1(𝑡),𝑛≥1. Lemma 2.2. The functions 𝐹^(0)𝑛,1(𝑡)=^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−1𝑑𝑡_𝑛𝑒^{−𝜇(𝑡−𝑡𝑛)𝑛}_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}(2.14) satisfy the difference-differential equations 𝑑²𝑑𝑡²𝐹^(0)𝑛,1(𝑡)=−𝜇𝑑𝑑𝑡𝐹^(0)𝑛,1(𝑡)+𝜇𝐹^{(0)𝑛−1,1}(𝑡),(2.15) where 𝑡₀=0, 𝑡>0,𝑛≥1.Proof. We first note that 𝑑𝑑𝑡𝐹^(0)𝑛,1(𝑡)=𝑑𝑑𝑡^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−1𝑑𝑡_𝑛𝑒^{−𝜇(𝑡−𝑡𝑛)𝑛}=_𝑡0_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}𝑑𝑡₁⋯^𝑡𝑡_𝑛−2𝑑𝑡_𝑛−11−𝑒^{−𝜇(𝑡−𝑡𝑛−1)}^𝑛−1−𝜇_𝑡0_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}𝑑𝑡₁⋯^𝑡𝑡_𝑛−1𝑑𝑡_𝑛𝑒^{−𝜇(𝑡−𝑡𝑛)𝑛}_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)},(2.16) and therefore 𝑑²𝑑𝑡²𝐹^(0)𝑛,1(𝑡)=𝜇^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−2𝑑𝑡_𝑛−1𝑒^{−𝜇(𝑡−𝑡𝑛−1)𝑛−1}−𝜇_𝑡0_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}𝑑𝑡₁⋯^𝑡𝑡_𝑛−2𝑑𝑡_𝑛−11−𝑒^{−𝜇(𝑡−𝑡𝑛−1)}^𝑛−1+𝜇₂_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−1𝑑𝑡_𝑛𝑒^{−𝜇(𝑡−𝑡𝑛)𝑛}=−𝜇𝑑_𝑖=11−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}𝑑𝑡𝐹^(0)𝑛,1(𝑡)+𝜇𝐹^{(0)𝑛−1,1}(𝑡).(2.17)

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

Theorem 2.3. The function 𝑓_𝑇(1)(𝑡;1) satisfies the second-order linear homogeneous differential equation 𝑑²𝑑𝑡²𝑓_𝑇(1)(𝑡;1)=−(2𝜆+𝜇)𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;1)−𝜆²𝑓_𝑇(1)(𝑡;1),(2.18) with the initial conditions 𝑓_𝑇(1)(0;1)=𝜆,𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;1)|_𝑡=0=−𝜆(𝜆+𝜇).(2.19) The explicit value of 𝑓_𝑇(1)(𝑡;1) is 𝑓_𝑇(1)(𝑡;1)=𝜆𝑒^{−(𝑡/2)(2𝜆+𝜇)}√𝜇(4𝜆+𝜇)−𝜇2√𝜇(4𝜆+𝜇)𝑒^{√(𝑡/2)𝜇(4𝜆+𝜇)}+√𝜇(4𝜆+𝜇)+𝜇2√𝜇(4𝜆+𝜇)𝑒^{√−(𝑡/2)𝜇(4𝜆+𝜇)}.(2.20)

Proof. From (2.13), it follows that 𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;1)=−𝜆𝑓_𝑇(1)(𝑡;1)+𝑒^{∞−𝜆𝑡}_𝑛=0𝜆^𝑛+1𝑑𝑑𝑡𝐹^(0)𝑛,1(𝑡),(2.21) and thus, in view of (2.17) and by letting 𝐹^(0)−1,1(𝑡)=0 , we have that 𝑑²𝑑𝑡²𝑓_𝑇(1)(𝑡;1)=−𝜆𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;1)−𝜆𝑒^{∞−𝜆𝑡}_𝑛=0𝜆^𝑛+1𝑑𝑑𝑡𝐹^(0)𝑛,1(𝑡)+𝑒^{∞−𝜆𝑡}_𝑛=0𝜆^𝑛+1𝑑²𝑑𝑡²=−2𝜆𝑑𝐹^(0)𝑛,1(𝑡)𝑑𝑡𝑓^𝑇(1)(𝑡;1)−𝜆²𝑓_𝑇(1)(𝑡;1)+𝜆𝜇𝑓_𝑇(1)(𝑡;1)−𝜇𝑑𝑑𝑡𝑓^𝑇(1)=−(2𝜆+𝜇)𝑑(𝑡;1)+𝜆𝑓_𝑇(1)(𝑡;1)𝑑𝑡𝑓^𝑇(1)(𝑡;1)−𝜆²𝑓_𝑇(1)(𝑡;1).(2.22) While the first condition is straightforward to verify, the second one needs some explanations: if we write 𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;1)||_𝑡=0=lim_Δ𝑡→0𝑓_𝑇(1)(Δ𝑡;1)−𝑓_𝑇(1)(0;1)Δ𝑡,(2.23) and observe that 𝑓_𝑇(1)(Δ𝑡;1)=𝑒^{−𝜆Δ𝑡}𝜆𝐹^(0)0,1(Δ𝑡)=𝜆𝑒^{−(𝜆+𝜇)Δ𝑡}=𝜆[1−(𝜆+𝜇)Δ𝑡]+𝑜(Δ𝑡),(2.24) by substituting (2.24) in (2.23) the second condition emerges.
The general solution to (2.22) has the form
𝑒^{−(𝑡/2)(2𝜆+𝜇)}𝐴𝑒^{√(𝑡/2)𝜇(4𝜆+𝜇)}+𝐵𝑒^{√−(𝑡/2)𝜇(4𝜆+𝜇)}.(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 𝑓_𝑇(𝑖)(𝑡;1) do not depend on t, but on the time interval [𝑡_𝑙𝑖−1,𝑡), in fact if 𝑡_𝑙𝑖−1=𝑠, we have that 𝑓_𝑇(𝑖)(𝑡;1,𝑠)=𝑒^{∞−𝜆(𝑡−𝑠)}_𝑛=0𝜆^𝑛+1𝐹^(𝑠)𝑛,1(𝑡)=𝑓_𝑇(𝑖)(𝑡−𝑠;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)}_𝑖∈𝑍⁺ is a renewal process with
𝑓𝑓_𝑇(1)(𝑡)=𝑓_𝑇(𝑖)(𝑡)if𝜈(0)=1,_𝑇(1)(𝑡)≠𝑓_𝑇(𝑖)(𝑡)if𝜈(0)=0.(2.27)

Remark 2.5. If 𝜆→0 we get 𝑓_𝑇(1)(𝑡;1)=0, because the 1th interloss time is greater than 𝑡, ∨𝑡>0, when nobody enters.
If 𝜇→0, we have that 𝑓_𝑇(1)(𝑡;1)=𝜆𝑒^−𝜆𝑡 because without exits and with the system busy at 𝑡=0, the 1th interloss time has the same distribution of the interarrival time.

Remark 2.6. The probability density function 𝑓_𝑇(1)(𝑡;1) can be expressed by the following hyperbolic functions: 𝑓_𝑇(1)(𝑡;1)=𝜆𝑒^{(−𝑡/2)(2𝜆+𝜇)}cosh𝑡√𝜇(4𝜆+𝜇)2−√𝜇(4𝜇𝜆+𝜇)sinh𝑡√𝜇(4𝜆+𝜇)2.(2.28)

Now, if we assume that the system is free at the starting point, the event “The 1th interloss time is 𝑡” is represented by Figure 2.

Figure 2 

The 1th interloss time is $t$ with $P\{\nu (0)=0\}=1.$

Let 𝑃^𝑛(0)(𝑡) be the conditional probability that the nth entered customer is the 1th lost at time t, when 𝜈(0)=0, 𝑃^𝑛(0)(𝑡)==𝑃𝑃𝑙𝑆₁₁(𝑡)=𝑛∣𝜈(0)=0<𝜏₍₂₎,…,𝑆_𝑛−2<𝜏_(𝑛−1),𝑆_𝑛−1>𝑡−𝑡_𝑛−1,(2.29) it can be computed conditionally by the (𝑛−1) moments when the served customers have arrived at the system 𝑃^𝑛(0)(𝑡)=(𝑛−1)!𝑡^𝑛−1^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−2𝑑𝑡_𝑛−1𝑒^{−𝜇(𝑡−𝑡𝑛−1)𝑛−1}_𝑖=21−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}=(𝑛𝑡−1_𝑛−1)!𝐹^{(0)𝑛−1,0}(𝑡),(2.30) where 𝐹^{(0)𝑛−1,0}(𝑡)=^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−2𝑑𝑡_𝑛−1𝑒^{−𝜇(𝑡−𝑡𝑛−1)𝑛−1}_𝑖=21−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}.(2.31) The function 𝑓_𝑇(1)(𝑡;0) can be computed as mean of convolution of 𝑙₁(𝑡) exponential probability density functions. So we have that 𝑓_𝑇(1)(𝑡;0)=^∞_𝑛=2𝑓_∑^𝑛𝑘=1_𝜏(𝑘)(𝑡)𝑃^𝑛(0)(𝑡)=𝑒^{∞−𝜆𝑡}_𝑛=2𝜆^𝑛𝐹^{(0)𝑛−1,0}(𝑡).(2.32)

Lemma 2.7. The functions 𝐹^(0)𝑛,0(𝑡)=^𝑡0𝑑𝑡₁⋯^𝑡𝑡_𝑛−1𝑑𝑡_𝑛𝑒^{−𝜇(𝑡−𝑡𝑛)𝑛}_𝑖=21−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}(2.33) satisfy the difference-differential equations 𝑑²𝑑𝑡²𝐹^(0)𝑛,0(𝑡)=−𝜇𝑑𝑑𝑡𝐹^(0)𝑛,0(𝑡)+𝜇𝐹^{(0)𝑛−1,0}(𝑡),(2.34) where 𝑡>0,𝑛≥1,𝐹^(0)0,0(𝑡)=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 𝑓_𝑇(1)(𝑡;0) satisfies the second-order linear homogeneous differential equation 𝑑²𝑑𝑡²𝑓_𝑇(1)(𝑡;0)=−(2𝜆+𝜇)𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;0)−𝜆²𝑓_𝑇(1)(𝑡;0),(2.35) with the initial conditions 𝑓_𝑇(1)(0;0)=0,𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;0)||_𝑡=0=𝜆².(2.36) The explicit value of 𝑓_𝑇(1)(𝑡;0) is 𝑓_𝑇(1)(𝑡;0)=𝜆²√𝜇(4𝜆+𝜇)𝑒^{−(𝑡/2)(2𝜆+𝜇)}𝑒^{√(𝑡/2)𝜇(4𝜆+𝜇)}−𝑒^{√−(𝑡/2)𝜇(4𝜆+𝜇)}.(2.37)

Proof. By substituting in (2.21) and (2.22) 𝐹^(0)𝑛,1(𝑡) with 𝐹^(0)𝑛,0(𝑡), (2.35) emerges. While the first condition is straightforward to verify, the second one needs some explanations: if we write 𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;0)||_𝑡=0=lim_Δ𝑡→0𝑓_𝑇(1)(Δ𝑡;0)−𝑓_𝑇(1)(0;0)Δ𝑡,(2.38) and observe that 𝑓_𝑇(1)(Δ𝑡;0)=𝑒^{−𝜆Δ𝑡}𝜆²𝑒^{−𝜇Δ𝑡}^0Δ𝑡𝑒^𝜇𝑡1𝑑𝑡₁=𝜆²Δ𝑡+𝑜(Δ𝑡),(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 𝜆→0, we get 𝑓_𝑇(1)(𝑡;0)=0 because the 1th interloss time is greater than 𝑡, ∨𝑡>0, when nobody enters.
If 𝜇→0, we have that 𝑓_𝑇(1)(𝑡;0)=𝜆²𝑡𝑒^−𝜆𝑡 because without exits and with the system free at 𝑡=0, the 1th interloss time is equal to the sum of 1th and 2th interarrival times.

Remark 2.10. The function 𝑓_𝑇(1)(𝑡;0) can be expressed by the following hyperbolic function: 𝑓_𝑇(1)(𝑡;0)=2𝜆²𝑒^{−(𝑡/2)(2𝜆+𝜇)}√𝜇(4𝜆+𝜇)sinh𝑡√𝜇(4𝜆+𝜇)2.(2.40)

Remark 2.11. The function 𝑓_𝑇(1)(𝑡;0) can also be computed as the convolution of the exponential density with rate 𝜆 and 𝑓_𝑇(1)(𝑡;1): 𝑓_𝑇(1)(𝑡;0)=^𝑡0𝜆𝑒^−𝜆𝑡1𝑓_𝑇(1)𝑡−𝑡₁;1𝑑𝑡₁.(2.41)

In fact, if we observe that 𝐹^(0)𝑛,0(𝑡)=^𝑡0𝑑𝑡₁^𝑡𝑡₁𝑑𝑡₂⋯^𝑡𝑡_𝑛−1𝑑𝑡_𝑛𝑒^{−𝜇(𝑡−𝑡𝑛)𝑛}_𝑖=21−𝑒^{−𝜇(𝑡𝑖−𝑡𝑖−1)}=^𝑡0𝑑𝑡₁𝐹^{(𝑡1)𝑛−1,1}(𝑡),(2.42) by (2.7), (2.32), and (2.42) we obtain that 𝑓_𝑇(1)(𝑡;0)=𝑒^{∞−𝜆𝑡}_𝑛=2𝜆^𝑛^𝑡0𝑑𝑡₁𝐹^{(0)𝑛−2,1}𝑡−𝑡₁=𝑓_𝜏(1)∗𝑓_𝑇(1)(𝑡;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 𝑓_𝑇(1)(𝑡;1)𝐹^∗𝑇₍₁₎(𝑠;1)=^∞0𝑒^−𝑠𝑡𝑓_𝑇(1)(𝑡;1)𝑑𝑡(3.1) satisfies the equation 𝑠²𝐹^∗𝑇₍₁₎(𝑠;1)−𝑠𝜆+𝜆(𝜆+𝜇)=−(2𝜆+𝜇)𝑠𝐹^∗𝑇₍₁₎(𝑠;1)+𝜆(2𝜆+𝜇)−𝜆²𝐹^∗𝑇₍₁₎(𝑠;1),(3.2) and its explicit value is 𝐹^∗𝑇₍₁₎(𝑠;1)=𝜆(𝜆+𝑠)(𝜆+𝑠)²+𝑠𝜇.(3.3)

Proof. By (2.18), (2.19), and the property ^∞0𝑑𝑡𝑒^−𝑠𝑡𝑑𝑑𝑡𝑓^𝑇(1)(𝑡;1)=𝑠𝐹^∗𝑇₍₁₎(𝑠;1)−𝑓_𝑇(1)(0;1),(3.4) the results (3.2) and (3.3) emerge.

Remark 3.2. The 1th conditional interloss mean time Θ¹⁽¹⁾=𝐸𝑇₍₁₎∣𝜈(0)=1(3.5) can be found by evaluating the derivative of (3.3) with respect to 𝑠 in 𝑠=0, as follows: −𝑑𝑑𝑠𝐹^∗𝑇(1)(𝑠;1)||_𝑠=0=𝜆+𝜇𝜆².(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 𝑓_𝑇(1)(𝑡;0)𝐹^∗𝑇₍₁₎(𝑠;0)=^∞0𝑒^−𝑠𝑡𝑓_𝑇(1)(𝑡;0)𝑑𝑡(3.7) satisfies the equation 𝑠²𝐹^∗𝑇₍₁₎(𝑠;0)−𝜆²=−(2𝜆+𝜇)𝑠𝐹^∗𝑇₍₁₎(𝑠;0)−𝜆²𝐹^∗𝑇₍₁₎(𝑠;0),(3.8) and its explicit value is 𝐹^∗𝑇₍₁₎(𝑠;0)=𝜆²(𝜆+𝑠)²+𝑠𝜇.(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⁾=𝐸𝑇₍₁₎∣𝜈(0)=0(3.10) can be found by evaluating the derivative of (3.9) with respect to 𝑠 in 𝑠=0, as follows: −𝑑𝑑𝑠𝐹^∗𝑇(1)(𝑠;0)||_𝑠=0=Θ¹⁽¹⁾+1𝜆.(3.11)

Remark 3.5. By (2.41), the result (3.9) can be obtained using (3.3), as follows: 𝐹^∗𝑇₍₁₎(𝑠;0)=𝜆𝜆+𝑠𝐹^∗𝑇(1)(𝑠;1).(3.12)

Now, let 𝑇_𝑛 be the time spent between 𝑡=0 and the 𝑛th loss, the conditional density functions 𝑓_𝑇𝑛(𝑡;𝑖)=𝑑𝑑𝑡𝑃𝑇^𝑛<𝑡;𝑖,(3.13) with 𝑖=0,1, can be evaluated by convolutions as follows: if 𝑖=0, we have that 𝑓_𝑇𝑛(𝑡;0)=𝑓_𝜏(1)∗𝑓_𝑇(1)(𝑡;1)∗⋯∗𝑓_𝑇(𝑛)(𝑡;1),(3.14) while if 𝑖=1, we have that 𝑓_𝑇𝑛(𝑡;1)=𝑓_𝑇(1)(𝑡;1)∗⋯∗𝑓_𝑇(𝑛)(𝑡;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, 𝐹^∗𝑇_𝑛(𝑠;0)=𝐹^∗𝜏_{(0)+∑𝑛𝑘=1𝑇(𝑘)}(𝑠;1)=𝜆𝜆+𝑠(𝜆𝜆+(𝜆𝑠)+²+𝑠)𝜇𝑠^𝑛,(3.16)(ii) if 𝑖=1, 𝐹^∗𝑇_𝑛(𝑠;1)=𝐹^∗∑_{𝑛𝑘=1𝑇(𝑘)}(𝑠;1)=𝜆(𝜆+𝑠)(𝜆+𝑠)²+𝜇𝑠^𝑛.(3.17)

Remark 3.6. The conditional averages Θ^𝑛(𝑖)=𝐸𝑇_𝑛∣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 −𝑑𝑑𝑠𝐹^∗𝑇𝑛(𝑠;0)||_𝑠=0=𝑛Θ¹⁽¹⁾+1𝜆,(3.19)(ii)if 𝑖=1, we have that −𝑑𝑑𝑠𝐹^∗𝑇𝑛(𝑠;1)||_𝑠=0=𝑛Θ¹⁽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.