IBM ( IBM) began a $340 million cash tender offer to acquire French software maker ILOG ( ILOG) Tuesday, a purchase the U.S. company hopes will boost its presence in the lucrative business process management market. The tech giant's Citloi subsidiary is offering 10 euros, or $13.70, for each of ILOG's shares in France and the company's American depositary shares held in the U.S. Both offers are expected to end Nov. 17. Shares in IBM, which recently issued an encouraging earnings report, were up 3.9% to $95.76, despite a broader slump in tech stocks. ILOG was gaining 3.1% to $13.40. BPM software is used to plan, monitor and analyze business processes such as billing, distribution and marketing, and it's seen as a way for firms to streamline their internal systems and improve customer service. Faced with a significant downturn in tech spending, users are under increasing pressure to wring as much value as possible out of their existing IT hardware. BPM has already been identified as one of the key technologies in a tight economic climate. Analyst firm IDC, for example, estimates that the BPM market is growing 44% each year and could be worth as much as $5.5 billion in 2011. The ILOG offer is IBM's latest foray into the world of BPM, and it follows the company's $1.6 billion acquisition of FileNet in 2006 and the purchase of privately held AptSoft for an undisclosed fee earlier this year. IBM is planning to build ILOG's software into its own WebSphere technology, and it has already highlighted the French company's ability to apply "business rules" to corporate computer systems. This could involve, for example, pushing a frequent customer to the front of a company's phone queue.