PEEE TRANSADPONS ON RELIABILITY, VOL. 45. NO. 1, 1996 MARCH

Comment/Reply : Some Recent Papers on Consecutively-Connected Systems

F. K. Hwang AT&T Bell Laboratories, Murray Hill

Key Words - Consecutively-connected system, multi-state system, algorithm.

1. INTRODUCTION

Acronyms

CCS (2-state) consecutively-connected system MCCS linear multi-state consecutively-connected system L/YY, C/YY [linear, circular]/YY Rel-XX reliability of XX.

Nomenclature & Notation

CCS. Shanthikumar [4] proposed the L/CCS where n 2-state components are arrayed in a line between the source (com- ponent #0) and the sink (component #[n+ l]), where compo- nent #i, i = 0, 1,. . . ,n is directly connected to the subsequent k,, k, 5 n + 1 -i components. Component i is good (work- ing) with probability p l and when good, can reach any node to which it is directly connected. A failed (non-good) com- ponent can not reach any node. po = pn+l = 1. The system is good (working) if the source has a direct path to the sink. MCCS. Hwang & Yao [l] extended L/CCS to L/MCCS in

Kossow & Preuss [2] and Malinowski & Preuss [3] pro- vided algorithms for Rel-L/MCCS and Rel-C/MCCS. Both papers made good technical points, but either were unaware of, or misunderstood, the proper references. The L/MCCS & C/MCCS problems were proposed in [l], and [l] was about the only published paper on MCCS before [2,3]. Therefore it seems that the only literature from which [2,3] could have picked-up the problem was [l]; [3] did refer to [l], but [2] did not.

Ref [3], in referring to [l], stated that [l] studied only CCS, not MCCS. However, in [l] - 1) the title begins with Multistate; 2) paragraph #1 ends with, “In this paper the com- ponents are not necessarily 2-state, viz, working-failed. ” ; 3) assumption #1 states: “Component i has probability p t j (. ..) to be directly connected to all the j - i subsequent components ... .”; and 4) the results correspond to MCCS.

Due to omitting and/or misquoting a very relevant reference, 1) the reader can misunderstand the history and the status of the problem; and 2) it is difficult for the future re- searcher to clarify the record. Refs [2,3] also lost the oppor- tunity to compare their algorithms with existing ones, and thus left open the problem of justifying the new proposals.

Jacek Malinowski

Wolfgang Preuss

Dresden

which component #i has 1 failure state and ki good (work- ing) states. Component i is good in state j , j = 1,2,. . . ,ki with probability pi,> In state j , component #i is directly connected to the subsequentj components. Thus L/CCS is a special case of L/MCCS where component i has 1 working state kj.

Polish Academy of Science, Warsaw

Hochschule fur Technik & Wirtschaft Dresden,

k = max{ki: O s i s n ] . 4

Standard notation is given in ‘‘Information for Readers & Authors” at the rear of each issue.

2. DISCUSSION

Shanthikumar [4] gave an O(n2)-time algorithm for Rel-L/CCS.

Hwang & Yao [l] gave an O(k.n) algorithm for Rel- L/MCCS; they extended L/MCCS to C/MCCS and gave an O(m-n2)-time algorithm for Rel-C/MCCS; m is the size of a typically small set.

Zuo [5] gave an 0 (k.n)-time algorithm for Rel-WCCS and an 0 ( k3 .n)-time algorithm for Rel-C/CCS. There was a slight oversight in [5] in that an existing O(k.n)-time algorithm for Rel-L/CCS existed as a special case of the O(k-n)-time algorithm for Rel-L/MCCS by Hwang & Yao [l].

Hwang & Yao [l] presented two algorithms for Rel-L/CCS & Rel-C/CCS. While the algorithm for Rel-L/CCS evaluates system reliability with multi-state components, the algorithm for Rel-C/CCS (being an extension of the Shanthikumar algorithm [4]) can be applied only to systems with 2-state com- ponents. The abstract [ 11 states, “ . . . , although for 2-state com- ponents we are able to extend the Shanthikumar algorithm to circular systems. ”

In [3] we restricted our attention to C/CCS and consequent- ly, our remark concerning [1,5] was limited to C/CCS. The Introduction [3] states that, circular CCS have been studied in [l] and [5] but only with 2-state components.

Another algorithm for Rel-C/MCCS was found in- dependently by Zuo & Liang [6].

(Continued on page 160)

0018-9629/96/$5.00 01996 IEEE