The impact of disruption characteristics on the performance of a server

Abstract

In this paper, we study a queueing system serving N customers with an unreliable server subject to disruptions even when idle. Times between server interruptions, service times, and times between customer arrivals are assumed to follow exponential distributions. The main contribution of the paper is to use general distributions for the length of server interruption periods/down times. Our numerical analysis reveals the importance of incorporating the down time distribution into the model, since their impact on customer service levels could be counterintuitive. For instance, while higher down time variability increases the mean queue length, for other service levels, can prove to be improving system performance. We also show how the process completion time approach from the literature can be extended to analyze the queueing system if the unreliable server fails only when it is serving a customer.

Appendices

Appendix 1: Proofs

Proof of Theorem 1

If we divide both sides of Eq. (7) by $e^{-{N\lambda y -\int _{0}^{y}\beta (x)dx}}P_{N,1}(0)$ and Eq. (8) by $e^{-{i\lambda y -\int _{0}^{y}\beta (x)dx}}P_{N,1}(0)$, we get

$$\begin{aligned} \frac{d}{dy}\left( \frac{e^{N\lambda y +\int _{0}^{y}\beta (x)dx}P_{N,1}(y)}{P_{N,1}(0)} \right)= & {} 0, \end{aligned}$$

(23)

$$\begin{aligned} \frac{d}{dy}\left( \frac{e^{i\lambda y +\int _{0}^{y}\beta (x)dx}P_{i,1}(y)}{P_{N,1}(0)} \right)= & {} \frac{(i+1)\lambda e^{i\lambda y +\int _{0}^{y}\beta (x)dx}P_{i+1,1}(y)}{P_{N,1}(0)},\nonumber \\&0\le i\le N-1, \end{aligned}$$

(24)

which are first order differential equations. We solve Eqs. (23) and (24) using Eq. (21) as

$$\begin{aligned} Q_{N}(y)= & {} 1, \end{aligned}$$

(25)

$$\begin{aligned} Q_{i}(y)= & {} Q_{i}(0)+(i+1)\lambda \int ^y_0{Q_{i+1}(x)e^{-\lambda x}dx,} \nonumber \\&0\le i\le N-1. \end{aligned}$$

(26)

Considering the definition given in Eq. (21), and employing Eqs. (5), (6) and (9), we obtain

$$\begin{aligned} Q_{N-1}(0)= & {} \frac{N\lambda +\alpha -\alpha \int ^{\infty }_0{Q_{N}(y)e^{-N\lambda y}f(y)}dy}{\mu }, \end{aligned}$$

(27)

$$\begin{aligned} Q_{i-1}(0)= & {} \frac{(i\lambda +\mu +\alpha )Q_{i}(0)-(i+1)\lambda Q_{i+1}(0)-\alpha \int ^{\infty }_0 Q_{i}(y)e^{-i\lambda y}f(y)dy}{\mu },\nonumber \\&1\le i\le N-1. \end{aligned}$$

(28)

For simplicity, we define

$$\begin{aligned} {\mathcal {B}}_i= & {} \int ^{\infty }_0Q_{i}(y)e^{-i\lambda y}f(y)dy. \end{aligned}$$

(29)

In order ${\mathcal {Q}}_i$ and ${\mathcal {B}}_i$ to be finite, we will show that $Q_{i}(y)$ is bounded for all $i=0,\ldots ,N$, which is proved in the following Lemma.

Lemma 1

$\lim _{y\rightarrow \infty } Q_{i}(y)=Q_{i}(\infty )$ exists and is finite. We also have $Q_{i}(y)\le Q_{i}(\infty )$.

Proof. Lemma 1

From Eq. (21), $Q_{i}(y)\ge 0$ and from Eq. (26), we see that $Q_{i}(y)$ is increasing in y. Let $Q_{i}(\infty )=\lim _{y\rightarrow \infty } Q_{i}(y)$. Then, $Q_{i}(y)\le Q_{i}(\infty ),\ 0\le i\le N-1$. If we take the limit as $y\rightarrow \infty$ in Eq. (26),

$$\begin{aligned} Q_{i}\left( \infty \right) \le Q_{i}\left( 0\right) +\left( i+1\right) Q_{i+1}\left( \infty \right) ,\quad 0\le i\le N-1. \end{aligned}$$

Starting with $Q_{N}(\infty )=1$ (due to Eq. 25) and using induction from the above equation, we see that $Q_{i}(\infty )$ is finite for all $i=0,\ldots ,N$. $\square$

Let $\Phi _{i}(s)=\int _{0}^{\infty }Q_{i}(y)e^{-sy}dy$ be the LT of the function $Q_{i}(y)$. In this case, the LT’s of $Q_{N}(y)$ and $Q_{i}(y)$ from Eqs. (25) and (26) will be

$$\begin{aligned} \Phi _{N}(s)= & {} \frac{1}{s}, \end{aligned}$$

(30)

$$\begin{aligned} \Phi _{i}(s)= & {} \frac{1}{s}{\mathcal {Q}}_{i}+(i+1)\frac{\lambda }{s}\Phi _{i+1}(\lambda +s), \quad 0\le i\le N-1. \end{aligned}$$

(31)

Starting from Eq. (30) and using the recursive formula in Eq. (31), we establish

$$\begin{aligned} \Phi _{i}(s)=\sum ^N_{j=i}{{j}\atopwithdelims (){i}}\frac{(j-i)!\lambda ^{(j-i)}}{s(\lambda +s)\cdots ((j-i)\lambda +s)}{\mathcal {Q}}_j, \quad 0\le i\le N-1. \end{aligned}$$

(32)

Using

$$\begin{aligned} \frac{k!{\lambda }^k}{s(\lambda +s)\cdots (k\lambda +s)}=\sum ^k_{j=0}{{(-1)}^j{{k}\atopwithdelims (){j}}\frac{1}{j\lambda +s}}, \end{aligned}$$

Eq. (32) can be rewritten as

$$\begin{aligned} \Phi _{i}(s)=\sum ^N_{j=i}{{j}\atopwithdelims (){i}}\mathcal Q_j\sum ^{j-i}_{l=0}(-1)^l{{j-i}\atopwithdelims (){l}}\frac{1}{l\lambda +s}, \quad 0\le i\le N-1. \end{aligned}$$

(33)

Observe that $(l\lambda +s)^{-1}$ on the right hand side of Eq. (33) is the LT of $e^{-l\lambda y}$. Using this, when we invert $\Phi _{i}(s)$, we obtain

$$\begin{aligned} Q_{i}(y)= & {} \sum ^N_{j=i}{{j}\atopwithdelims (){i}}{\mathcal {Q}}_j\sum ^{j-i}_{l=0}(-1)^l{{j-i}\atopwithdelims (){l}}e^{-l\lambda y}\nonumber \\= & {} \sum ^N_{j=i}{{j}\atopwithdelims (){i}}{\mathcal {Q}}_j\sum ^{j-i}_{l=0}{{j-i}\atopwithdelims (){l}}(-e^{-\lambda y})^l\nonumber \\= & {} \sum ^N_{j=i}{{{j}\atopwithdelims (){i}}{\mathcal {Q}}_j{(1-e^{-\lambda y})}^{j-i}}, \quad 0\le i\le N-1. \end{aligned}$$

(34)

Substituting Eq. (34) in Eq. (29), we have Eq. (14) where

$$\begin{aligned} \zeta _{i,j}={{j}\atopwithdelims (){i}} \int ^{\infty }_0(1-e^{-\lambda y})^{j-i}e^{-i\lambda y}f(y)dy, \quad j \ge i. \end{aligned}$$

(35)

This leads to Eqs. (12) and (13). Together with Eq. (14) as defined in Eq. (29), Eq. (22) gives Eqs. (10) and (11).

We define ${\mathcal {D}}_i=i\lambda \int ^{\infty }_0(P_{i,1}(y)/P_{N,1}(0))dy$. Noting from Eq. (26) that $dQ_{i}(y)=(i+1)\lambda Q_{i+1}(y)e^{-\lambda y}$, if we rewrite Eq. (29) as ${\mathcal {B}}_i=-\int ^{\infty }_0Q_{i}(y)e^{-i\lambda y}d\overline{F}(y)$, integration yields

$$\begin{aligned} {\mathcal {B}}_i= & {} Q_{i}(0)+(i+1)\lambda \int ^{\infty }_0Q_{i+1}(y)e^{-(i+1)\lambda y}\overline{F}(y)dy-i\lambda \int ^{\infty }_0Q_{i}(y)e^{-i\lambda y}\overline{F}(y)dy. \end{aligned}$$

Considering Eq. (21) for ${\mathcal {D}}_i$, the above given equation gives Eqs. (15) and (18). With Eq. (9) and the definitions given in Eqs. (21) and (22), we obtain Eq. (17). $\square$

Proof. Corollary 1

Substituting Eq. (21) in Eq. (34), we arrive at Eq. (19)

Proof of Theorem 2

By definition $\sum ^N_{i=0}\overline{P}_i=\sum ^N_{i=0}(P_{i,0}+\int ^{\infty }_0P_{i,1}(y)dy)=1$, which by using Eq. (9), becomes $\sum ^N_{i=0}(P_{i,0}(0)/\alpha +\int ^{\infty }_0P_{i,1}(y)dy)=1$. If we divide this equation by $P_{N,1}(0)$, we have

$$\begin{aligned} P_{N,1}(0)=\left( \frac{1}{\alpha }S_N(0)+\int ^{\infty }_0{S_N(y)}dy\right) ^{-1}, \end{aligned}$$

(36)

where $S_N(y)=\sum ^N_{i=0}P_{i,1}(y)/P_{N,1}(0)$.

Summing up Eqs. (7) and (8), we obtain the first order differential equation

$$\begin{aligned} \frac{d}{dy}S_N(y)=-\beta (y)S_N(y), \end{aligned}$$

that has a solution of $S_N(y)=S_N(0)e^{-\int ^y_0{\beta (x)dx}}=S_N(0)\overline{F}(y)$. Substituting this in Eq. (36) and using the fact that $E[D]=\int _0^{\infty }\overline{F}(y)dy$ gives us

$$\begin{aligned} P_{N,1}(0)=\left( S_N(0)\left( \frac{1}{\alpha }+E[D]\right) \right) ^{-1}. \end{aligned}$$

Considering Eqs. (22) and (21), $S_N(0)=\sum ^N_{i=0}{\mathcal {Q}}_i$; after substituting it in the equation given above, and using the boundary condition in Eq. (9) we obtain Eq. (20). $\square$

Appendix 2: The ODD queue

In this section, we summarize how results from the literature can be easily used in analyzing the queue where unlike the model studied in this paper, the server can experience disruptions only if it has customers. We make the same assumptions and employ the same notations introduced in Sect. 2 for the underlying r.v.s with two differences. First, the actual service time of a job—in the absence of disruptions—is a general i.i.d. r.v. with the LT of $\tilde{b}(s)$. Second, while $\alpha$ still denotes the rate of the exponential times to interruption, the interruption process is halted when the server becomes idle until it becomes busy again. To handle this problem, one can use the process completion time (PCT) r.v. (Gaver 1962) that is the total time a customer spends on the server including its actual service time plus possible OFF periods it may experience. Let C denote the PCT r.v. the LT of which is given by (e.g., Altiok 1997, p. y94)

$$\begin{aligned} \tilde{c}(s)=\tilde{b}\left( s+\alpha -\alpha \tilde{f}(s)\right) , \end{aligned}$$

(37)

where $\tilde{f}(s)$ is the LT of the OFF periods.

The PCT r.v., C, includes all the information of ON and OFF periods. We can use it as the service time r.v. in an queue without interruptions analyzed by Gupta and Srinivasa Rao (1996), which will be referred to as the queue. Note that the queue has the same $\lambda$ and N as in the original ODD queue, and additionally uses $\tilde{c}(s)$ from Eq. (37) as the LT of the service time. Using the algorithm by Gupta and Srinivasa Rao, one obtains $P_i$ in the queue, which coincides with the probability of having i customers in the original ODD queue. With these probabilities, the expected system size and the probability of the server is idle in the ODD queue can be computed. The limitation of using the PCT approach is that we are unable to find the probability that the server is down or serving a customer, $P_D$ and $P_B$, respectively.