Analysis of overtaking in M/M/c queues Woo-Sung Kim a , Dae-Eun Lim b,⇑ a The School of Management and Economics, Handong Global University, Republic of Korea b Department of System & Management Engineering, Kangwon National University, Republic of Korea article info Article history: Received 16 March 2015 Received in revised form 18 August 2016 Accepted 2 September 2016 Available online 7 September 2016 Keywords: Overtaking M/M/c queue Parallel processing Resequencing abstract Overtaking events can occur in parallel processing systems when customers are served simultaneously by multiple servers. We say that one customer overtakes another when the customer leaves the system after having been served ahead of another customer(s) who arrived earlier. Overtaking events are an important issue in flexible assembly systems or packet switched communication networks. Although materials (or packets) can be processed by different servers simultaneously, there is a designated order for the assem- bly of final products (or data). If overtaking occurs, then additional time and money may be required to rearrange materials (or packets) to the original order (called resequencing). We investigate overtaking in an M/M/c queueing system. Two distributions were considered to describe the amount of overtaking: the number of customers that an arbitrary (tagged) customer overtakes and the number of customers who overtake the tagged customer. Explicit forms of these distributions are provided for some cases. Finally, we apply our results to practical issues in flexible assembly systems and telecommunication systems. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction Customer A is said to overtake customer B when A arrives later than B, but departs earlier. Overtaking is commonly observed: for example, when going through the departure process at an airport, someone who was initially behind you in the queue may finish the check-in process earlier at another counter. Overtaking in this kind of service system is directly related to fairness. However, when overtaking events occur in telecommunication systems or produc- tion systems, shuffled packets or jobs must be resequenced (i.e., unscrambled) according to the original sequence. In many flexible assembly systems, additional spaces called buffers can be desig- nated after the resequencing process. This means that additional time and space (cost) resources are required. Moreover, overtaking may have a negative impact on the performance of communication networks that require real-time transmissions. Although overtak- ing can be highly influential on a system’s performance or reputa- tion, only a few studies have examined this phenomenon in detail. The goal of this paper is to define measures for the amount of overtaking and to obtain useful results. Specifically, we derive the explicit probability distribution of the number of customers that an arbitrary (tagged) customer overtakes, and the probability that the tagged customer is overtaken by the next customer. More- over, we present the expected number of customers that will over- take a tagged customer. The expected number can be obtained explicitly in the form of a generating function for the M/M/2 queue. In single-server queueing systems, only employed service poli- cies can influence the number of overtaking events. No overtaking can happen under the First-Come-First-Served (FCFS) approach, whereas the maximum number of overtaking events occurs under the Last-Come-First-Served (LCFS) protocol. The relationship between the service discipline and the variance in the sojourn times of customers for single-server queues was studied by Kingman (1962, 1982). Because overtaking events influence sojourn times (Whitt, 1984), it is of interest to study the amount of overtaking according to different service disciplines for single- server queues. In this paper, we consider a multi-server M/M/c queueing system, which is more applicable to real-world applica- tions than single-server models. The waiting time for a standard M/M/c queue is less than that of an M/M/1 system for any c > 1 given that the arrival and service rates are identical; this was pro- vided by Medhi (2002) as an exercise problem (see 3.1(b), p. 145) based on the work of Rothkopf and Rech (1987) and Smith and Whitt (1981). The FCFS discipline was also assumed. There have been numerous results on the M/M/c queue for var- ious aspects, from classical books such as Medhi (2002) and Gross, Shortle, Thompson, and Harris (2013) to transient characteristics (Al-Seedy, El-Sherbiny, El-Shehawy, & Ammar, 2009). For more http://dx.doi.org/10.1016/j.cie.2016.09.005 0360-8352/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: wskim@handong.edu (W.-S. Kim), del@kangwon.ac.kr (D.-E. Lim). Computers & Industrial Engineering 101 (2016) 177–183 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie