本文主要討論關於一位不可靠的服務員工作被中斷後，立刻重新工作的等待系統。針對此系統我們給定工作所需要的全部時間之變異數為常數來刻劃指數分佈。此結果可以視為一種指數分佈之無記憶性質。

Consider a single-server queueing system with an unreliable server and service repetition. In this system, if the service is interrupted, the service is restarted anew immediately and ends whenever the service period is failure free. In this paper, we give some characterization of the
exponential distribution by the constancy of total service time in this system. The result can be viewed as a kind of memoryless property of the exponential distribution.

Contents
1. Introduction..................................1
2. Preliminaries.................................3
3. Characterization based on variance............5
References.......................................7

1 Apostol, T. M.(1974).
Mathematical Analysis, 2nd ed. Addison-Wesley, Reading, Massachusetts.
2 Azlarov, T.A. and Volodin, N.A. (1986).
Characterization Problems Associated with the Exponential
Distribution (Translated by Stein, M. and Edited by Olkin,
I.). Springer-Verlag, New York.
3 Dimitrov, B. and Khalil, Z. (1990). On a new
characterization of the exponential distribution related to a
queueing system with an unreliable server.
J. Appl. Prob. 27, 221-226.
4 Galambos, J. and Kotz, S. (1978).
Characterizations of Probability Distributions. { Lecture Notes in
Mathematics} {f 675}, Springer-Verlag, New York.
5 Huang, W.J. and Shoung, J.M. (1993). Some
characterizations of the exponential and geometric
distributions arising in a queueing system with an unreliable
server. {it J. Appl. Prob.} {f 30}, 985-990.
6 Huang, W.J., Li, S.H. and Su,J.C. (1993). Some
characterizations of the Poisson process and geometric renewal
process. {it J. Appl. Prob.} {f 30}, 121-130.
7 Huang, W.J. and Chen, W.P. (1996). Some characterizations of
the geometric distribution related to a single-server queueing
system with an unreliable server. (in Chinese) {it J.C.S.A.} {f 34}, 275-287.
8 Huang, W.J. and Chang, W.C. (2000). On a study
of the exponential and Poisson characteristics of the Poisson
process. {it Metrika} {f 50}, 247-254.
9 Khalil, Z. and Dimitrov, B. (1994) The service
time properties of an unreliable server characterize the
exponential distribution. {it Adv. Appl. Prob.} {f 26}, 172-182.
10 Khalil, Z., Dimitrov, B. and Dion, J.P. (1991)
A characterization of the geometric
distribution related to random sums. {it Commun. Statist.-Stoch. Models} {f 7}, 321-326.
11 Lin, G.D. (1993). Characterizations of the exponential distribution
via the blocking time in a queueing system. {it Statistica Sinica} {f 3}, 577-581.
12 Ross, S.L. (1984). Differential Equations, 3rd ed. Wiley, New York.
13 Silverman, R.A. (1974). Complex Analysis with
Applications. Pren-tice-Hall, Inc., Englewood Cliffs, N. J.
14 Simmons, G.F. (1991). Differential Equations with
Applications and Historical Notes, 2nd ed. McGraw-Hill, New York.
15 Van Harn, K. and Steutel, F.W. (1991). On a characterization of the
exponential distribution. {it J. Appl. Prob.} {f 28}, 947-949.
16 Xu, J.L. (1998) Characterization of the exponential distribution
via the blocking time in a queueing system with an unreliable
server. {it J. Appl. Prob.} {f 35}, 236-239.